5.3 Trigonometric Functions of Any Angle
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Section 5.3 Trigonometric Functions of Any Angle 104. From the top of a 250-foot lighthouse, a plane is sighted overhead and a ship is observed directly below the plane. The angle of elevation of the plane is 22° and the angle of depression of the ship is 35°. Find a. the distance of the ship from the lighthouse; b. the plane’s height above the water. Round to the nearest foot.
Preview Exercises Exercises 105–107 will help you prepare for the material covered in the next section. Use these figures to solve Exercises 105–106. y
105. a. Write a ratio that expresses sin u for the right triangle in Figure (a). b. Determine the ratio that you wrote in part (a) for Figure (b) with x = - 3 and y = 4. Is this ratio positive or negative? 106. a. Write a ratio that expresses cos u for the right triangle in Figure (a). b. Determine the ratio that you wrote in part (a) for Figure (b) with x = - 3 and y = 5. Is this ratio positive or negative? 107. Find the positive angle u¿ formed by the terminal side of u and the x-axis. a.
y P ⫽ (x, y)
r
y
b.
u
u x
y
x
x
5p
u⬘
Section
5.3
Objectives Use the definitions of
u⫽ 6
x
(b) u lies in quadrant II.
(a) u lies in quadrant I.
x
u⬘
y
x
y
u ⫽ 345⬚
P ⫽ (x, y) r
trigonometric functions of any angle. Use the signs of the trigonometric functions. Find reference angles. Use reference angles to evaluate trigonometric functions.
513
Trigonometric Functions of Any Angle
T
here is something comforting in the repetition of some of nature’s patterns. The ocean level at a beach varies between high and low tide approximately every 12 hours. The number of hours of daylight oscillates from a maximum on the summer solstice, June 21, to a minimum on the winter solstice, December 21. Then it increases to the same maximum the following June 21. Some believe that cycles, called biorhythms, represent physical, emotional, and intellectual aspects of our lives. Throughout the remainder of this chapter, we will see how the trigonometric functions are used to model phenomena that occur again and again. To do this, we need to move beyond right triangles.
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514 Chapter 5 Trigonometric Functions
Trigonometric Functions of Any Angle
Use the definitions of trigonometric functions of any angle.
In the last section, we evaluated trigonometric functions of acute angles, such as that shown in Figure 5.32(a). Note that this angle is in standard position. The point P = 1x, y2 is a point r units from the origin on the terminal side of u. A right triangle is formed by drawing a line segment from P = 1x, y2 perpendicular to the x-axis. Note that y is the length of the side opposite u and x is the length of the side adjacent to u.
y
y
y
P (x, y)
u x
x
P (x, y) r y
y
r
y x
(a) u lies in quadrant I.
Figure 5.32
Study Tip If u is acute, we have the right triangle shown in Figure 5.32(a). In this situation, the definitions in the box are the right triangle definitions of the trigonometric functions. This should make it easier for you to remember the six definitions.
x
x r
r u
u x
y
u
y P (x, y)
x
x
P (x, y)
(b) u lies in quadrant II.
(c) u lies in quadrant III.
(d) u lies in quadrant IV.
Figures 5.32(b), (c), and (d) show angles in standard position, but they are not acute. We can extend our definitions of the six trigonometric functions to include such angles, as well as quadrantal angles. (Recall that a quadrantal angle has its terminal side on the x-axis or y-axis; such angles are not shown in Figure 5.32.) The point P = 1x, y2 may be any point on the terminal side of the angle u other than the origin, (0, 0).
Definitions of Trigonometric Functions of Any Angle
Let u be any angle in standard position and let P = 1x, y2 be a point on the
terminal side of u. If r = 3x2 + y2 is the distance from (0, 0) to 1x, y2, as shown in Figure 5.32, the six trigonometric functions of U are defined by the following ratios: y r x cos u= r y tan u= , x 0 x sin u=
r ,y0 y r sec u= , x 0 x x cot u= , y 0. y csc u=
The ratios in the second column are the reciprocals of the corresponding ratios in the first column.
Because the point P = 1x, y2 is any point on the terminal side of u other than the origin, (0, 0), r = 3x2 + y2 cannot be zero. Examine the six trigonometric functions defined above. Note that the denominator of the sine and cosine functions is r. Because r Z 0, the sine and cosine functions are defined for any angle u. This is not true for the other four trigonometric functions. Note that the y r denominator of the tangent and secant functions is x: tan u = and sec u = . x x These functions are not defined if x = 0. If the point P = 1x, y2 is on the y-axis, then x = 0. Thus, the tangent and secant functions are undefined for all quadrantal angles with terminal sides on the positive or negative y-axis. Likewise, if P = 1x, y2 is on the x-axis, then y = 0, and the cotangent and cosecant functions are x r undefined: cot u = and csc u = . The cotangent and cosecant functions y y are undefined for all quadrantal angles with terminal sides on the positive or negative x-axis.
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Evaluating Trigonometric Functions
EXAMPLE 1
Let P = 1- 3, -52 be a point on the terminal side of u. Find each of the six trigonometric functions of u. y
Solution The situation is shown in Figure 5.33. We need values for x, y, and r to evaluate all six trigonometric functions. We are given the values of x and y. Because P = 1- 3, - 52 is a point on the terminal side of u, x = - 3 and y = - 5. Furthermore,
5
u −5
5 r
Now that we know x, y, and r, we can find the six trigonometric functions of u. Where appropriate, we will rationalize denominators.
−5 P = (−3, − 5) x = −3
r = 3x2 + y2 = 31-322 + 1- 522 = 29 + 25 = 234.
x
sin u =
y -5 5 # 234 5234 = = = r 34 234 234 234
csc u =
r 234 234 = = y -5 5
cos u =
x -3 3 # 234 3234 = = = r 34 234 234 234
sec u =
r 234 234 = = x -3 3
tan u =
y -5 5 = = x -3 3
cot u =
x -3 3 = = y -5 5
y = −5
Figure 5.33
Let P = 11, - 32 be a point on the terminal side of u. Find each of the six trigonometric functions of u.
Check Point
1
How do we find the values of the trigonometric functions for a quadrantal angle? First, draw the angle in standard position. Second, choose a point P on the angle’s terminal side. The trigonometric function values of u depend only on the size of u and not on the distance of point P from the origin. Thus, we will choose a point that is 1 unit from the origin. Finally, apply the definitions of the appropriate trigonometric functions.
EXAMPLE 2
Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the sine function and the tangent function at the following four quadrantal angles: p 3p a. u = 0° = 0 b. u = 90° = c. u = 180° = p d. u = 270° = . 2 2
Solution y u = 0 −1
x=1
y=0
P = (1, 0) 1
x
a. If u = 0° = 0 radians, then the terminal side of the angle is on the positive x-axis. Let us select the point P = 11, 02 with x = 1 and y = 0. This point is 1 unit from the origin, so r = 1. Figure 5.34 shows values of x, y, and r corresponding to u = 0° or 0 radians. Now that we know x, y, and r, we can apply the definitions of the sine and tangent functions.
r=1
Figure 5.34
sin 0° = sin 0 =
y 0 = = 0 r 1
tan 0° = tan 0 =
y 0 = = 0 x 1
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516 Chapter 5 Trigonometric Functions p radians, then the terminal side of the angle is on the positive 2 y-axis. Let us select the point P = 10, 12 with x = 0 and y = 1. This point is 1 unit from the origin, so r = 1. Figure 5.35 shows values of x, y, and r p corresponding to u = 90° or . Now that we know x, y, and r, we can apply the 2 definitions of the sine and tangent functions.
y
b. If u = 90° = x=0
y=1
P = (0, 1)
1 r=1
u = 90
−1
x
1
y p 1 = = = 1 r 2 1 y 1 p = tan 90° = tan = x 2 0 sin 90° = sin
Figure 5.35
Because division by 0 is undefined, tan 90° is undefined. c. If u = 180° = p radians, then the terminal side of the angle is on the negative x-axis. Let us select the point P = 1-1, 02 with x = - 1 and y = 0. This point is 1 unit from the origin, so r = 1. Figure 5.36 shows values of x, y, and r corresponding to u = 180° or p. Now that we know x, y, and r, we can apply the definitions of the sine and tangent functions.
y x = −1
y=0
1 u = 180
P = (−1, 0) −1
1
x
sin 180° = sin p =
y 0 = = 0 r 1
tan 180° = tan p =
y 0 = = 0 x -1
r=1
Figure 5.36
u = 270 −1 r=1
x
1 −1
3p radians, then the terminal side of the angle is on the negative 2 y-axis. Let us select the point P = 10, - 12 with x = 0 and y = - 1. This point is 1 unit from the origin, so r = 1. Figure 5.37 shows values of x, y, and r 3p corresponding to u = 270° or . Now that we know x, y, and r, we can apply 2 the definitions of the sine and tangent functions.
d. If u = 270° =
y
P = (0, −1) x=0
y 3p -1 = = = -1 r 2 1 y -1 3p tan 270° = tan = = x 2 0 sin 270° = sin
y = −1
Figure 5.37
Discovery Try finding tan 90° and tan 270° with your calculator. Describe what occurs.
Because division by 0 is undefined, tan 270° is undefined.
2
Check Point Evaluate, if possible, the cosine function and the cosecant function at the following four quadrantal angles: p 3p a. u = 0° = 0 b. u = 90° = c. u = 180° = p d. u = 270° = . 2 2 Use the signs of the trigonometric functions.
The Signs of the Trigonometric Functions
y Quadrant II sine and cosecant positive
Quadrant I All functions positive x
Quadrant III tangent and cotangent positive
Quadrant IV cosine and secant positive
Figure 5.38 The signs of the trigonometric functions
In Example 2, we evaluated trigonometric functions of quadrantal angles. However, we will now return to the trigonometric functions of nonquadrantal angles. If U is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which U lies. In all four quadrants, r is positive. However, x and y can be positive or negative. For example, if u lies in quadrant II, x is negative and y is positive. Thus, the y r only positive ratios in this quadrant are and its reciprocal, . These ratios are the r y function values for the sine and cosecant, respectively. In short, if u lies in quadrant II, sin u and csc u are positive. The other four trigonometric functions are negative. Figure 5.38 summarizes the signs of the trigonometric functions. If u lies in quadrant I, all six functions are positive. If u lies in quadrant II, only sin u and csc u are positive. If u lies in quadrant III, only tan u and cot u are positive. Finally, if u lies in quadrant IV, only cos u and sec u are positive. Observe that the positive functions in each quadrant occur in reciprocal pairs.
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517
Study Tip Here’s a phrase to help you remember the signs of the trig functions:
A
Smart
All trig functions are positive in QI.
Sine and its reciprocal, cosecant, are positive in QII.
Trig
Class.
Tangent and its reciprocal, cotangent, are positive in QIII.
Cosine and its reciprocal, secant, are positive in QIV.
Finding the Quadrant in Which an Angle Lies
EXAMPLE 3
If tan u 6 0 and cos u 7 0, name the quadrant in which angle u lies.
Solution When tan u 6 0, u lies in quadrant II or IV. When cos u 7 0, u lies in quadrant I or IV. When both conditions are met (tan u 6 0 and cos u 7 0), u must lie in quadrant IV.
Check Point
3
If sin u 6 0 and cos u 6 0, name the quadrant in which angle
u lies.
Evaluating Trigonometric Functions
EXAMPLE 4
Given tan u = - 23 and cos u 7 0, find cos u and csc u.
Solution Because the tangent is negative and the cosine is positive, u lies in quadrant IV. This will help us to determine whether the negative sign in tan u = - 23 should be associated with the numerator or the denominator. Keep in mind that in quadrant IV, x is positive and y is negative. Thus,
y 5
r = 兹13
u −5
5
−5
P = (3, − 2) x=3
y –2 2 tan u=– = = . x 3 3 (See Figure 5.39.) Thus, x = 3 and y = - 2. Furthermore, r = 3x2 + y2 = 332 + 1- 222 = 29 + 4 = 213.
y = −2
Figure 5.39 tan u = - 23 and cos u 7 0
In quadrant IV, y is negative. x
Now that we know x, y, and r, we can find cos u and csc u. cos u =
x 3 3 # 213 3213 = = = r 13 213 213 213
Check Point
4
csc u =
r 213 213 = = y -2 2
Given tan u = - 13 and cos u 6 0, find sin u and sec u.
In Example 4, we used the quadrant in which u lies to determine whether a negative sign should be associated with the numerator or the denominator. Here’s a situation, similar to Example 4, where negative signs should be associated with both the numerator and the denominator: tan u =
3 5
and cos u 6 0.
Because the tangent is positive and the cosine is negative, u lies in quadrant III. In quadrant III, x is negative and y is negative. Thus, y 3 –3 tan u= = = . x 5 –5
We see that x = −5 and y = −3.
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518 Chapter 5 Trigonometric Functions
Find reference angles.
Reference Angles We will often evaluate trigonometric functions of positive angles greater than 90° and all negative angles by making use of a positive acute angle. This positive acute angle is called a reference angle.
Definition of a Reference Angle Let u be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle u¿ formed by the terminal side of u and the x-axis. Figure 5.40 shows the reference angle for u lying in quadrants II, III, and IV. Notice that the formula used to find u¿, the reference angle, varies according to the quadrant in which u lies. You may find it easier to find the reference angle for a given angle by making a figure that shows the angle in standard position. The acute angle formed by the terminal side of this angle and the x-axis is the reference angle. y
y
u
y
u
u
x
u
x
x
u
u
Figure 5.40 Reference angles, u¿, for positive angles, u, in quadrants II, III, and IV
If 90 u 180, then u 180 u.
EXAMPLE 5
If 180 u 270, then u u 180.
If 270 u 360, then u 360 u.
Finding Reference Angles
Find the reference angle, u¿, for each of the following angles: a. u = 345°
Solution
b. u =
5p 6
c. u = - 135°
d. u = 2.5. y
a. A 345° angle in standard position is shown in Figure 5.41. Because 345° lies in quadrant IV, the reference angle is u¿ = 360° - 345° = 15°. p 5p 3p = lies between and 6 2 6 6p 5p p = ,u = lies in quadrant II. The 6 6 angle is shown in Figure 5.42. The reference angle is
u 345
x
u 15
Figure 5.41
b. Because
u¿ = p -
Discovery Solve part (c) by first finding a positive coterminal angle for - 135° less than 360°. Use the positive coterminal angle to find the reference angle.
6p 5p p 5p = = . 6 6 6 6
c. A -135° angle in standard position is shown in Figure 5.43. The figure indicates that the positive acute angle formed by the terminal side of u and the x-axis is 45°. The reference angle is u¿ = 45°.
y 5p
u 6
p
u 6
x
Figure 5.42 y
u 45
Figure 5.43
x u 135
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Section 5.3 Trigonometric Functions of Any Angle
p L 1.57 and p L 3.14. This means that 2 u = 2.5 is in quadrant II, shown in Figure 5.44. The reference angle is
y
u 艐 0.64
519
d. The angle u = 2.5 lies between u 2.5 x
u¿ = p - 2.5 L 0.64. Figure 5.44
5
Check Point a. u = 210°
Find the reference angle, u¿, for each of the following angles: b. u =
7p 4
c. u = - 240°
d. u = 3.6.
Finding reference angles for angles that are greater than 360° 12p2 or less than - 360° 1- 2p2 involves using coterminal angles. We have seen that coterminal angles have the same initial and terminal sides. Recall that coterminal angles can be obtained by increasing or decreasing an angle’s measure by an integer multiple of 360° or 2p.
Finding Reference Angles for Angles Greater Than 360° (2P) or Less Than 360° (2P) 1. Find a positive angle a less than 360° or 2p that is coterminal with the given angle. 2. Draw a in standard position. 3. Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side of a and the x-axis is the reference angle.
EXAMPLE 6
Finding Reference Angles
Find the reference angle for each of the following angles: 8p 13p a. u = 580° b. u = c. u = . 3 6
Solution a. For a 580° angle, subtract 360° to find a positive coterminal angle less than 360°. y
580° - 360° = 220° Figure 5.45 shows a = 220° in standard position. Because 220° lies in quadrant III, the reference angle is a¿ = 220° - 180° = 40°.
a 220 a 40
x
Figure 5.45
8p 2 b. For an , or 2 p, angle, subtract 2p to find a positive coterminal angle less 3 3 than 2p. 8p 6p 2p 8p - 2p = = 3 3 3 3 y 2p Figure 5.46 shows a = in standard position. 3 2p 2p Because lies in quadrant II, the reference p a 3 a 3 3 x angle is 2p 3p 2p p a¿ = p = = . Figure 5.46 3 3 3 3
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520 Chapter 5 Trigonometric Functions Discovery Solve part (c) using the coterminal angle formed by adding 2p, rather than 4p, to the given angle.
y
13p 1 , or -2 p, angle, add 4p to find a 6 6 positive coterminal angle less than 2p.
c. For a -
11p
a 6
p
a 6
13p 13p 24p 11p + 4p = + = 6 6 6 6
x
Figure 5.47
11p in standard position. 6 11p Because lies in quadrant IV, the reference angle is 6 Figure 5.47 shows a =
a¿ = 2p -
Check Point a. u = 665°
Use reference angles to evaluate trigonometric functions.
6
11p 12p 11p p = = . 6 6 6 6
Find the reference angle for each of the following angles: b. u =
15p 4
c. u = -
11p . 3
Evaluating Trigonometric Functions Using Reference Angles The way that reference angles are defined makes them useful in evaluating trigonometric functions.
Using Reference Angles to Evaluate Trigonometric Functions The values of the trigonometric functions of a given angle, u, are the same as the values of the trigonometric functions of the reference angle, u¿, except possibly for the sign. A function value of the acute reference angle, u¿, is always positive. However, the same function value for u may be positive or negative. For example, we can use a reference angle, u¿, to obtain an exact value for tan 120°. The reference angle for u = 120° is u¿ = 180° - 120° = 60°. We know the exact value of the tangent function of the reference angle: tan 60° = 23. We also know that the value of a trigonometric function of a given angle, u, is the same as that of its reference angle, u¿, except possibly for the sign. Thus, we can conclude that tan 120° equals - 23 or 23. What sign should we attach to 23? A 120° angle lies in quadrant II, where only the sine and cosecant are positive. Thus, the tangent function is negative for a 120° angle. Therefore, Prefix by a negative sign to show tangent is negative in quadrant II.
tan 120=–tan 60=–兹3. The reference angle for 120° is 60°.
In the previous section, we used two right triangles to find exact trigonometric values of 30°, 45°, and 60°. Using a procedure similar to finding tan 120°, we can now find the exact function values of all angles for which 30°, 45°, or 60° are reference angles.
A Procedure for Using Reference Angles to Evaluate Trigonometric Functions The value of a trigonometric function of any angle u is found as follows: 1. Find the associated reference angle, u¿, and the function value for u¿. 2. Use the quadrant in which u lies to prefix the appropriate sign to the function value in step 1.
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Discovery
EXAMPLE 7
Draw the two right triangles involving 30°, 45°, and 60°. Indicate the length of each side. Use these lengths to verify the function values for the reference angles in the solution to Example 7.
521
Using Reference Angles to Evaluate Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric functions: 4p p a. sin 135° b. cos c. cot a - b. 3 3
Solution a. We use our two-step procedure to find sin 135°. Step 1 Find the reference angle, U œ , and sin U œ . Figure 5.48 shows 135° lies in quadrant II. The reference angle is
y
u¿ = 180° - 135° = 45°. 135 45 x
The function value for the reference angle is sin 45° =
22 . 2
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the function value in step 1. The angle u = 135° lies in quadrant II. Because the sine is positive in quadrant II, we put a + sign before the function value of the reference angle. Thus,
Figure 5.48 Reference angle for 135°
The sine is positive in quadrant II.
sin 135=±sin 45=
兹2 . 2
The reference angle for 135° is 45°.
b. We use our two-step procedure to find cos y 4p 3
x p 3
4p . 3
Step 1 Find the reference angle, U œ , and cos U œ . Figure 5.49 shows that 4p u = lies in quadrant III. The reference angle is 3 4p 3p p 4p - p = = . u¿ = 3 3 3 3 The function value for the reference angle is
Figure 5.49 Reference
cos
4p angle for 3
p 1 = . 3 2
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the 4p function value in step 1. The angle u = lies in quadrant III. Because only the 3 tangent and cotangent are positive in quadrant III, the cosine is negative in this quadrant. We put a - sign before the function value of the reference angle. Thus, The cosine is negative in quadrant III.
cos
1 p 4p =–cos =– . 2 3 3 The reference angle for 4p is p. 3 3
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522 Chapter 5 Trigonometric Functions c. We use our two-step procedure to find cot a -
Step 1 Find the reference angle, U œ , and cot U œ . Figure 5.50 shows that p p u = - lies in quadrant IV. The reference angle is u¿ = . The function value 3 3 p 23 for the reference angle is cot = . 3 3
y
x
p 3
p 3
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the p function value in step 1. The angle u = - lies in quadrant IV. Because only the 3 cosine and secant are positive in quadrant IV, the cotangent is negative in this quadrant. We put a - sign before the function value of the reference angle. Thus,
Figure 5.50 Reference angle for -
p b. 3
p 3
The cotangent is negative in quadrant IV.
p p 兹3 cot a– b =–cot =– . 3 3 3 The reference angle for − p is p. 3 3
7
Check Point Use reference angles to find the exact value of the following trigonometric functions: a. sin 300°
b. tan
c. sec a -
5p 4
p b. 6
In our final example, we use positive coterminal angles less than 2p to find the reference angles.
EXAMPLE 8
Using Reference Angles to Evaluate Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric functions: a. tan
14p 3
b. sec a -
17p b. 4
Solution 14p . 3 14p Step 1 Find the reference angle, U œ , and tan U œ . Because the given angle, 3 2 or 4 p, exceeds 2p, subtract 4p to find a positive coterminal angle less than 2p. 3 14p 14p 12p 2p u = - 4p = = 3 3 3 3
a. We use our two-step procedure to find tan
y
p 3
2p 3
x
Figure 5.51 Reference angle for
2p 3
2p Figure 5.51 shows u = in standard position. The angle lies in quadrant II. 3 The reference angle is 2p 3p 2p p u¿ = p = = . 3 3 3 3 p = 23. The function value for the reference angle is tan 3
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Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the 2p function value in step 1. The coterminal angle u = lies in quadrant II. 3 Because the tangent is negative in quadrant II, we put a - sign before the function value of the reference angle. Thus, The tangent is negative in quadrant II.
tan
p 2p 14p =tan =–tan =–兹3 . 3 3 3 The reference angle for 2p is p. 3 3
b. We use our two-step procedure to find sec a -
17p b. 4
Step 1 Find the reference angle, U œ , and sec U œ . Because the given angle, 17p 1 or -4 p, is less than -2p, add 6p (three multiples of 2p) to find a 4 4 positive coterminal angle less than 2p. u = -
Figure 5.52 shows u =
y
17p 17p 24p 7p + 6p = + = 4 4 4 4 7p in standard position. The angle lies in quadrant IV. 4
The reference angle is 7p 4 p 4
Figure 5.52 Reference angle for
7p 4
x
u¿ = 2p -
8p 7p p 7p = = . 4 4 4 4
The function value for the reference angle is sec
p = 22. 4
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the 7p function value in step 1. The coterminal angle u = lies in quadrant IV. 4 Because the secant is positive in quadrant IV, we put a + sign before the function value of the reference angle. Thus, The secant is positive in quadrant IV.
sec a–
17p 7p p b =sec =± sec =兹2 . 4 4 4 The reference angle for 7p is p. 4 4
Check Point
8
Use reference angles to find the exact value of each of the following trigonometric functions:
a. cos
17p 6
b. sin a -
22p b. 3
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524 Chapter 5 Trigonometric Functions Study Tip Evaluating trigonometric functions like those in Example 8 and Check Point 8 involves using a number of concepts, including finding coterminal angles and reference angles, locating special angles, determining the signs of trigonometric functions in specific quadrants, p p p and finding the trigonometric functions of special angles a 30° = , 45° = , and 60° = b. To be successful in trigonometry, it is 6 4 3 often necessary to connect concepts. Here’s an early reference sheet showing some of the concepts you should have at your fingertips (or memorized).
Degree and Radian Measures of Special and Quadrantal Angles p
90 , 2
2p
120 , 3 3p 135 , 4 5p
150 , 6
7p
11p
7p
330 , 6 7p 315 , 4 4p
240 , 3
3p
270 , 2
3p
270 , 2
210 , 6
0 , 0
210 , 6 5p 225 , 4
5p
300 , 3
180 , p
45
30°
U
1
30
P 6
5p
150 , 6 3p 135 , 4 2p 120 , 3
P 4
45°
p
30 , 6 p 45 , 4 p 60 , p 3
90 , 2
Signs of the Trigonometric Functions
60°
sin U
1 2
22 2
23 2
cos U
23 2
22 2
1 2
tan U
23 3
1
23
兹3
60
5p
300 , 3 7p 315 , 4 11p 330 , 6 0 , 0
Special Right Triangles and Trigonometric Functions of Special Angles
1
2
p
30 , 6
180 , p
兹2
4p
240 , 3 5p 225 , 4
p
60 , 3 p 45 , 4
P 3
y Quadrant II sine and cosecant positive
Quadrant I All functions positive x
Quadrant III tangent and cotangent positive
Quadrant IV cosine and secant positive
1
Using Reference Angles to Evaluate Trigonometric Functions
Trigonometric Functions of Quadrantal Angles U sin U cos U tan U
0° 0 0 1 0
90°
P 2
1 0 undefined
180° P 0 -1 0
270°
3P 2
-1 0 undefined
sin u= cos u= tan u=
sin u cos u tan u
+ or − in determined by the quadrant in which u lies and the sign of the function in that quadrant.
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525
Exercise Set 5.3 Practice Exercises In Exercises 1–8, a point on the terminal side of angle u is given. Find the exact value of each of the six trigonometric functions of u. 1. 1 -4, 32
2. 1- 12, 52
4. (3, 7)
7. 1- 2, -52
3. (2, 3)
5. 13, - 32
6. 15, -52
8. 1 - 1, -32
In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. 9. cos p
10. tan p 3p 13. tan 2 p 16. tan 2
12. csc p p 15. cot 2
11. sec p 3p 14. cos 2
In Exercises 17–22, let u be an angle in standard position. Name the quadrant in which u lies. 17. sin u 7 0, cos u 7 0
18. sin u 6 0, cos u 7 0
19. sin u 6 0, cos u 6 0
20. tan u 6 0, sin u 6 0
21. tan u 6 0, cos u 6 0
22. cot u 7 0, sec u 6 0
In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of u. 23. cos u = - 35 ,
u in quadrant III
24. sin u = - 12 13 ,
u in quadrant III
25. sin u = 26. cos u = 27. cos u = 28. cos u = 29. tan u = 31. tan u =
5 13 , u in quadrant II 4 5 , u in quadrant IV 8 17 , 270° 6 u 6 360° 1 3 , 270° 6 u 6 360° - 23 , sin u 7 0 30. 4 32. 3 , cos u 6 0
33. sec u = - 3, tan u 7 0
tan u = - 13 , tan u =
5 12 ,
2p 3 7p 70. cot 4
7p 6 9p 72. tan 2
3p 4 9p 71. tan 4
67. sin
68. cos
69. csc
73. sin1 - 240°2
74. sin1 -225°2
75. tan a -
76. tan a -
77. sec 495°
78. sec 510°
p b 6 19p 79. cot 6 35p 82. cos 6 17p b 85. sin a 3
13p 3 17p b 83. tan a 6 35p b 86. sin a 6 80. cot
p b 4
23p 4 11p b 84. tan a 4 81. cos
Practice Plus In Exercises 87–92, find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. p p 3p cos p - cos sin 3 3 2 p p 88. sin cos 0 - sin cos p 4 6 11p 5p 11p cos + cos sin 89. sin 4 6 4 17p 5p 17p cos + cos sin 90. sin 3 4 3 87. sin
5p 6 5p 4
91. sin
3p 15p 5p tan a b - cos a b 2 4 3
92. sin
3p 8p 5p tan a b + cos a b 2 3 6
sin u 7 0 cos u 6 0
In Exercises 93–98, let f1x2 = sin x, g1x2 = cos x, and h1x2 = 2x.
34. csc u = - 4, tan u 7 0
Find the exact value of each expression. Do not use a calculator. In Exercises 35–60, find the reference angle for each angle. 35. 160° 38. 210° 7p 41. 4 5p 44. 7 47. - 335° 50. 5.5 17p 53. 6 17p 56. 3 25p 59. 6
36. 170° 39. 355° 5p 42. 4
37. 205° 40. 351° 5p 43. 6
45. - 150°
46. -250°
48. -359° 51. 565° 11p 54. 4 11p 57. 4 13p 60. 3
49. 4.7 52. 553° 23p 55. 4 17p 58. 6
93. f a
4p p 4p p + b + fa b + fa b 3 6 3 6
94. g a
5p p 5p p + b + ga b + ga b 6 6 6 6
95. 1h ⴰ g2a
96. 1h ⴰ f2a
11p b 4
97. the average rate of change of f from x1 = 98. the average rate of change of g from x1 =
5p 3p to x2 = 4 2
3p to x2 = p 4
In Exercises 99–104, find two values of u, 0 … u 6 2p, that satisfy each equation. 99. sin u =
In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.
17p b 3
22 2
61. cos 225°
62. sin 300°
63. tan 210°
22 2
64. sec 240°
65. tan 420°
66. tan 405°
103. tan u = - 23
101. sin u = -
100. cos u =
1 2
102. cos u = -
1 2
104. tan u = -
23 3
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526 Chapter 5 Trigonometric Functions
Writing in Mathematics 105. If you are given a point on the terminal side of angle u, explain how to find sin u. 106. Explain why tan 90° is undefined.
113. When I found the exact value of cos 14p 3 , I used a number of concepts, including coterminal angles, reference angles, finding the cosine of a special angle, and knowing the cosine’s sign in various quadrants.
107. If cos u 7 0 and tan u 6 0, explain how to find the quadrant in which u lies. 108. What is a reference angle? Give an example with your description. 109. Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.
Critical Thinking Exercises Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning. 110. I’m working with a quadrantal angle u for which sin u is undefined. 111. This angle u is in a quadrant in which sin u 6 0 and csc u 7 0. 112. I am given that tan u = 35 , so I can conclude that y = 3 and x = 5.
Section
5.4
Objectives
Preview Exercises Exercises 114–116 will help you prepare for the material covered in the next section. 114. Graph: x2 + y2 = 1. Then locate the point A - 12 , graph.
B on the
115. Use your graph of x2 + y2 = 1 from Exercise 114 to determine the relation’s domain and range. 116. a. Find the exact value of sin
A p4 B , sin A - p4 B , sin A p3 B , and
sin A - B . Based on your results, can the sine function be an even function? Explain your answer. p 3
b. Find the exact value of cos cos A -
p 3
A p4 B , cos A - p4 B , cos A p3 B , and
B . Based on your results, can the cosine
function be an odd function? Explain your answer.
Trigonometric Functions of Real Numbers; Periodic Functions
Use a unit circle to define
13 2
C
ycles govern many aspects of life—heartbeats, sleep patterns, seasons, and tides all follow regular, predictable cycles. In this section, we will see why trigonometric functions are used to model phenomena that occur in cycles. To do this, we need to move beyond angles and consider trigonometric functions of real numbers.
trigonometric functions of real numbers. Recognize the domain and range of sine and cosine functions. Use even and odd trigonometric functions. Use periodic properties.
Use a unit circle to define trigonometric functions of real numbers.
Trigonometric Functions of Real Numbers
y
1
s t x (1, 0)
x2 + y2 = 1
Figure 5.53 Unit circle with a central angle measuring t radians
Thus far, we have considered trigonometric functions of angles measured in degrees or radians. To define trigonometric functions of real numbers, rather than angles, we use a unit circle. A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is x2 + y2 = 1. Figure 5.53 shows a unit circle in which the central angle measures t radians. We can use the formula for the length of a circular arc, s = ru, to find the length of the intercepted arc. s=ru=1 ⴢ t=t The radius of a unit circle is 1.
The radian measure of the central angle is t.
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Section 5.3 Trigonometric Functions of Any Angle 104. From the top of a 250-foot lighthouse, a plane is sighted overhead and a ship is observed directly below the plane. The angle of elevation of the plane is 22° and the angle of depression of the ship is 35°. Find a. the distance of the ship from the lighthouse; b. the plane’s height above the water. Round to the nearest foot.
Preview Exercises Exercises 105–107 will help you prepare for the material covered in the next section. Use these figures to solve Exercises 105–106. y
105. a. Write a ratio that expresses sin u for the right triangle in Figure (a). b. Determine the ratio that you wrote in part (a) for Figure (b) with x = - 3 and y = 4. Is this ratio positive or negative? 106. a. Write a ratio that expresses cos u for the right triangle in Figure (a). b. Determine the ratio that you wrote in part (a) for Figure (b) with x = - 3 and y = 5. Is this ratio positive or negative? 107. Find the positive angle u¿ formed by the terminal side of u and the x-axis. a.
y P ⫽ (x, y)
r
y
b.
u
u x
y
x
x
5p
u⬘
Section
5.3
Objectives Use the definitions of
u⫽ 6
x
(b) u lies in quadrant II.
(a) u lies in quadrant I.
x
u⬘
y
x
y
u ⫽ 345⬚
P ⫽ (x, y) r
trigonometric functions of any angle. Use the signs of the trigonometric functions. Find reference angles. Use reference angles to evaluate trigonometric functions.
513
Trigonometric Functions of Any Angle
T
here is something comforting in the repetition of some of nature’s patterns. The ocean level at a beach varies between high and low tide approximately every 12 hours. The number of hours of daylight oscillates from a maximum on the summer solstice, June 21, to a minimum on the winter solstice, December 21. Then it increases to the same maximum the following June 21. Some believe that cycles, called biorhythms, represent physical, emotional, and intellectual aspects of our lives. Throughout the remainder of this chapter, we will see how the trigonometric functions are used to model phenomena that occur again and again. To do this, we need to move beyond right triangles.
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514 Chapter 5 Trigonometric Functions
Trigonometric Functions of Any Angle
Use the definitions of trigonometric functions of any angle.
In the last section, we evaluated trigonometric functions of acute angles, such as that shown in Figure 5.32(a). Note that this angle is in standard position. The point P = 1x, y2 is a point r units from the origin on the terminal side of u. A right triangle is formed by drawing a line segment from P = 1x, y2 perpendicular to the x-axis. Note that y is the length of the side opposite u and x is the length of the side adjacent to u.
y
y
y
P (x, y)
u x
x
P (x, y) r y
y
r
y x
(a) u lies in quadrant I.
Figure 5.32
Study Tip If u is acute, we have the right triangle shown in Figure 5.32(a). In this situation, the definitions in the box are the right triangle definitions of the trigonometric functions. This should make it easier for you to remember the six definitions.
x
x r
r u
u x
y
u
y P (x, y)
x
x
P (x, y)
(b) u lies in quadrant II.
(c) u lies in quadrant III.
(d) u lies in quadrant IV.
Figures 5.32(b), (c), and (d) show angles in standard position, but they are not acute. We can extend our definitions of the six trigonometric functions to include such angles, as well as quadrantal angles. (Recall that a quadrantal angle has its terminal side on the x-axis or y-axis; such angles are not shown in Figure 5.32.) The point P = 1x, y2 may be any point on the terminal side of the angle u other than the origin, (0, 0).
Definitions of Trigonometric Functions of Any Angle
Let u be any angle in standard position and let P = 1x, y2 be a point on the
terminal side of u. If r = 3x2 + y2 is the distance from (0, 0) to 1x, y2, as shown in Figure 5.32, the six trigonometric functions of U are defined by the following ratios: y r x cos u= r y tan u= , x 0 x sin u=
r ,y0 y r sec u= , x 0 x x cot u= , y 0. y csc u=
The ratios in the second column are the reciprocals of the corresponding ratios in the first column.
Because the point P = 1x, y2 is any point on the terminal side of u other than the origin, (0, 0), r = 3x2 + y2 cannot be zero. Examine the six trigonometric functions defined above. Note that the denominator of the sine and cosine functions is r. Because r Z 0, the sine and cosine functions are defined for any angle u. This is not true for the other four trigonometric functions. Note that the y r denominator of the tangent and secant functions is x: tan u = and sec u = . x x These functions are not defined if x = 0. If the point P = 1x, y2 is on the y-axis, then x = 0. Thus, the tangent and secant functions are undefined for all quadrantal angles with terminal sides on the positive or negative y-axis. Likewise, if P = 1x, y2 is on the x-axis, then y = 0, and the cotangent and cosecant functions are x r undefined: cot u = and csc u = . The cotangent and cosecant functions y y are undefined for all quadrantal angles with terminal sides on the positive or negative x-axis.
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Section 5.3 Trigonometric Functions of Any Angle
515
Evaluating Trigonometric Functions
EXAMPLE 1
Let P = 1- 3, -52 be a point on the terminal side of u. Find each of the six trigonometric functions of u. y
Solution The situation is shown in Figure 5.33. We need values for x, y, and r to evaluate all six trigonometric functions. We are given the values of x and y. Because P = 1- 3, - 52 is a point on the terminal side of u, x = - 3 and y = - 5. Furthermore,
5
u −5
5 r
Now that we know x, y, and r, we can find the six trigonometric functions of u. Where appropriate, we will rationalize denominators.
−5 P = (−3, − 5) x = −3
r = 3x2 + y2 = 31-322 + 1- 522 = 29 + 25 = 234.
x
sin u =
y -5 5 # 234 5234 = = = r 34 234 234 234
csc u =
r 234 234 = = y -5 5
cos u =
x -3 3 # 234 3234 = = = r 34 234 234 234
sec u =
r 234 234 = = x -3 3
tan u =
y -5 5 = = x -3 3
cot u =
x -3 3 = = y -5 5
y = −5
Figure 5.33
Let P = 11, - 32 be a point on the terminal side of u. Find each of the six trigonometric functions of u.
Check Point
1
How do we find the values of the trigonometric functions for a quadrantal angle? First, draw the angle in standard position. Second, choose a point P on the angle’s terminal side. The trigonometric function values of u depend only on the size of u and not on the distance of point P from the origin. Thus, we will choose a point that is 1 unit from the origin. Finally, apply the definitions of the appropriate trigonometric functions.
EXAMPLE 2
Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the sine function and the tangent function at the following four quadrantal angles: p 3p a. u = 0° = 0 b. u = 90° = c. u = 180° = p d. u = 270° = . 2 2
Solution y u = 0 −1
x=1
y=0
P = (1, 0) 1
x
a. If u = 0° = 0 radians, then the terminal side of the angle is on the positive x-axis. Let us select the point P = 11, 02 with x = 1 and y = 0. This point is 1 unit from the origin, so r = 1. Figure 5.34 shows values of x, y, and r corresponding to u = 0° or 0 radians. Now that we know x, y, and r, we can apply the definitions of the sine and tangent functions.
r=1
Figure 5.34
sin 0° = sin 0 =
y 0 = = 0 r 1
tan 0° = tan 0 =
y 0 = = 0 x 1
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516 Chapter 5 Trigonometric Functions p radians, then the terminal side of the angle is on the positive 2 y-axis. Let us select the point P = 10, 12 with x = 0 and y = 1. This point is 1 unit from the origin, so r = 1. Figure 5.35 shows values of x, y, and r p corresponding to u = 90° or . Now that we know x, y, and r, we can apply the 2 definitions of the sine and tangent functions.
y
b. If u = 90° = x=0
y=1
P = (0, 1)
1 r=1
u = 90
−1
x
1
y p 1 = = = 1 r 2 1 y 1 p = tan 90° = tan = x 2 0 sin 90° = sin
Figure 5.35
Because division by 0 is undefined, tan 90° is undefined. c. If u = 180° = p radians, then the terminal side of the angle is on the negative x-axis. Let us select the point P = 1-1, 02 with x = - 1 and y = 0. This point is 1 unit from the origin, so r = 1. Figure 5.36 shows values of x, y, and r corresponding to u = 180° or p. Now that we know x, y, and r, we can apply the definitions of the sine and tangent functions.
y x = −1
y=0
1 u = 180
P = (−1, 0) −1
1
x
sin 180° = sin p =
y 0 = = 0 r 1
tan 180° = tan p =
y 0 = = 0 x -1
r=1
Figure 5.36
u = 270 −1 r=1
x
1 −1
3p radians, then the terminal side of the angle is on the negative 2 y-axis. Let us select the point P = 10, - 12 with x = 0 and y = - 1. This point is 1 unit from the origin, so r = 1. Figure 5.37 shows values of x, y, and r 3p corresponding to u = 270° or . Now that we know x, y, and r, we can apply 2 the definitions of the sine and tangent functions.
d. If u = 270° =
y
P = (0, −1) x=0
y 3p -1 = = = -1 r 2 1 y -1 3p tan 270° = tan = = x 2 0 sin 270° = sin
y = −1
Figure 5.37
Discovery Try finding tan 90° and tan 270° with your calculator. Describe what occurs.
Because division by 0 is undefined, tan 270° is undefined.
2
Check Point Evaluate, if possible, the cosine function and the cosecant function at the following four quadrantal angles: p 3p a. u = 0° = 0 b. u = 90° = c. u = 180° = p d. u = 270° = . 2 2 Use the signs of the trigonometric functions.
The Signs of the Trigonometric Functions
y Quadrant II sine and cosecant positive
Quadrant I All functions positive x
Quadrant III tangent and cotangent positive
Quadrant IV cosine and secant positive
Figure 5.38 The signs of the trigonometric functions
In Example 2, we evaluated trigonometric functions of quadrantal angles. However, we will now return to the trigonometric functions of nonquadrantal angles. If U is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which U lies. In all four quadrants, r is positive. However, x and y can be positive or negative. For example, if u lies in quadrant II, x is negative and y is positive. Thus, the y r only positive ratios in this quadrant are and its reciprocal, . These ratios are the r y function values for the sine and cosecant, respectively. In short, if u lies in quadrant II, sin u and csc u are positive. The other four trigonometric functions are negative. Figure 5.38 summarizes the signs of the trigonometric functions. If u lies in quadrant I, all six functions are positive. If u lies in quadrant II, only sin u and csc u are positive. If u lies in quadrant III, only tan u and cot u are positive. Finally, if u lies in quadrant IV, only cos u and sec u are positive. Observe that the positive functions in each quadrant occur in reciprocal pairs.
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Section 5.3 Trigonometric Functions of Any Angle
517
Study Tip Here’s a phrase to help you remember the signs of the trig functions:
A
Smart
All trig functions are positive in QI.
Sine and its reciprocal, cosecant, are positive in QII.
Trig
Class.
Tangent and its reciprocal, cotangent, are positive in QIII.
Cosine and its reciprocal, secant, are positive in QIV.
Finding the Quadrant in Which an Angle Lies
EXAMPLE 3
If tan u 6 0 and cos u 7 0, name the quadrant in which angle u lies.
Solution When tan u 6 0, u lies in quadrant II or IV. When cos u 7 0, u lies in quadrant I or IV. When both conditions are met (tan u 6 0 and cos u 7 0), u must lie in quadrant IV.
Check Point
3
If sin u 6 0 and cos u 6 0, name the quadrant in which angle
u lies.
Evaluating Trigonometric Functions
EXAMPLE 4
Given tan u = - 23 and cos u 7 0, find cos u and csc u.
Solution Because the tangent is negative and the cosine is positive, u lies in quadrant IV. This will help us to determine whether the negative sign in tan u = - 23 should be associated with the numerator or the denominator. Keep in mind that in quadrant IV, x is positive and y is negative. Thus,
y 5
r = 兹13
u −5
5
−5
P = (3, − 2) x=3
y –2 2 tan u=– = = . x 3 3 (See Figure 5.39.) Thus, x = 3 and y = - 2. Furthermore, r = 3x2 + y2 = 332 + 1- 222 = 29 + 4 = 213.
y = −2
Figure 5.39 tan u = - 23 and cos u 7 0
In quadrant IV, y is negative. x
Now that we know x, y, and r, we can find cos u and csc u. cos u =
x 3 3 # 213 3213 = = = r 13 213 213 213
Check Point
4
csc u =
r 213 213 = = y -2 2
Given tan u = - 13 and cos u 6 0, find sin u and sec u.
In Example 4, we used the quadrant in which u lies to determine whether a negative sign should be associated with the numerator or the denominator. Here’s a situation, similar to Example 4, where negative signs should be associated with both the numerator and the denominator: tan u =
3 5
and cos u 6 0.
Because the tangent is positive and the cosine is negative, u lies in quadrant III. In quadrant III, x is negative and y is negative. Thus, y 3 –3 tan u= = = . x 5 –5
We see that x = −5 and y = −3.
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518 Chapter 5 Trigonometric Functions
Find reference angles.
Reference Angles We will often evaluate trigonometric functions of positive angles greater than 90° and all negative angles by making use of a positive acute angle. This positive acute angle is called a reference angle.
Definition of a Reference Angle Let u be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle u¿ formed by the terminal side of u and the x-axis. Figure 5.40 shows the reference angle for u lying in quadrants II, III, and IV. Notice that the formula used to find u¿, the reference angle, varies according to the quadrant in which u lies. You may find it easier to find the reference angle for a given angle by making a figure that shows the angle in standard position. The acute angle formed by the terminal side of this angle and the x-axis is the reference angle. y
y
u
y
u
u
x
u
x
x
u
u
Figure 5.40 Reference angles, u¿, for positive angles, u, in quadrants II, III, and IV
If 90 u 180, then u 180 u.
EXAMPLE 5
If 180 u 270, then u u 180.
If 270 u 360, then u 360 u.
Finding Reference Angles
Find the reference angle, u¿, for each of the following angles: a. u = 345°
Solution
b. u =
5p 6
c. u = - 135°
d. u = 2.5. y
a. A 345° angle in standard position is shown in Figure 5.41. Because 345° lies in quadrant IV, the reference angle is u¿ = 360° - 345° = 15°. p 5p 3p = lies between and 6 2 6 6p 5p p = ,u = lies in quadrant II. The 6 6 angle is shown in Figure 5.42. The reference angle is
u 345
x
u 15
Figure 5.41
b. Because
u¿ = p -
Discovery Solve part (c) by first finding a positive coterminal angle for - 135° less than 360°. Use the positive coterminal angle to find the reference angle.
6p 5p p 5p = = . 6 6 6 6
c. A -135° angle in standard position is shown in Figure 5.43. The figure indicates that the positive acute angle formed by the terminal side of u and the x-axis is 45°. The reference angle is u¿ = 45°.
y 5p
u 6
p
u 6
x
Figure 5.42 y
u 45
Figure 5.43
x u 135
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Section 5.3 Trigonometric Functions of Any Angle
p L 1.57 and p L 3.14. This means that 2 u = 2.5 is in quadrant II, shown in Figure 5.44. The reference angle is
y
u 艐 0.64
519
d. The angle u = 2.5 lies between u 2.5 x
u¿ = p - 2.5 L 0.64. Figure 5.44
5
Check Point a. u = 210°
Find the reference angle, u¿, for each of the following angles: b. u =
7p 4
c. u = - 240°
d. u = 3.6.
Finding reference angles for angles that are greater than 360° 12p2 or less than - 360° 1- 2p2 involves using coterminal angles. We have seen that coterminal angles have the same initial and terminal sides. Recall that coterminal angles can be obtained by increasing or decreasing an angle’s measure by an integer multiple of 360° or 2p.
Finding Reference Angles for Angles Greater Than 360° (2P) or Less Than 360° (2P) 1. Find a positive angle a less than 360° or 2p that is coterminal with the given angle. 2. Draw a in standard position. 3. Use the drawing to find the reference angle for the given angle. The positive acute angle formed by the terminal side of a and the x-axis is the reference angle.
EXAMPLE 6
Finding Reference Angles
Find the reference angle for each of the following angles: 8p 13p a. u = 580° b. u = c. u = . 3 6
Solution a. For a 580° angle, subtract 360° to find a positive coterminal angle less than 360°. y
580° - 360° = 220° Figure 5.45 shows a = 220° in standard position. Because 220° lies in quadrant III, the reference angle is a¿ = 220° - 180° = 40°.
a 220 a 40
x
Figure 5.45
8p 2 b. For an , or 2 p, angle, subtract 2p to find a positive coterminal angle less 3 3 than 2p. 8p 6p 2p 8p - 2p = = 3 3 3 3 y 2p Figure 5.46 shows a = in standard position. 3 2p 2p Because lies in quadrant II, the reference p a 3 a 3 3 x angle is 2p 3p 2p p a¿ = p = = . Figure 5.46 3 3 3 3
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520 Chapter 5 Trigonometric Functions Discovery Solve part (c) using the coterminal angle formed by adding 2p, rather than 4p, to the given angle.
y
13p 1 , or -2 p, angle, add 4p to find a 6 6 positive coterminal angle less than 2p.
c. For a -
11p
a 6
p
a 6
13p 13p 24p 11p + 4p = + = 6 6 6 6
x
Figure 5.47
11p in standard position. 6 11p Because lies in quadrant IV, the reference angle is 6 Figure 5.47 shows a =
a¿ = 2p -
Check Point a. u = 665°
Use reference angles to evaluate trigonometric functions.
6
11p 12p 11p p = = . 6 6 6 6
Find the reference angle for each of the following angles: b. u =
15p 4
c. u = -
11p . 3
Evaluating Trigonometric Functions Using Reference Angles The way that reference angles are defined makes them useful in evaluating trigonometric functions.
Using Reference Angles to Evaluate Trigonometric Functions The values of the trigonometric functions of a given angle, u, are the same as the values of the trigonometric functions of the reference angle, u¿, except possibly for the sign. A function value of the acute reference angle, u¿, is always positive. However, the same function value for u may be positive or negative. For example, we can use a reference angle, u¿, to obtain an exact value for tan 120°. The reference angle for u = 120° is u¿ = 180° - 120° = 60°. We know the exact value of the tangent function of the reference angle: tan 60° = 23. We also know that the value of a trigonometric function of a given angle, u, is the same as that of its reference angle, u¿, except possibly for the sign. Thus, we can conclude that tan 120° equals - 23 or 23. What sign should we attach to 23? A 120° angle lies in quadrant II, where only the sine and cosecant are positive. Thus, the tangent function is negative for a 120° angle. Therefore, Prefix by a negative sign to show tangent is negative in quadrant II.
tan 120=–tan 60=–兹3. The reference angle for 120° is 60°.
In the previous section, we used two right triangles to find exact trigonometric values of 30°, 45°, and 60°. Using a procedure similar to finding tan 120°, we can now find the exact function values of all angles for which 30°, 45°, or 60° are reference angles.
A Procedure for Using Reference Angles to Evaluate Trigonometric Functions The value of a trigonometric function of any angle u is found as follows: 1. Find the associated reference angle, u¿, and the function value for u¿. 2. Use the quadrant in which u lies to prefix the appropriate sign to the function value in step 1.
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Section 5.3 Trigonometric Functions of Any Angle
Discovery
EXAMPLE 7
Draw the two right triangles involving 30°, 45°, and 60°. Indicate the length of each side. Use these lengths to verify the function values for the reference angles in the solution to Example 7.
521
Using Reference Angles to Evaluate Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric functions: 4p p a. sin 135° b. cos c. cot a - b. 3 3
Solution a. We use our two-step procedure to find sin 135°. Step 1 Find the reference angle, U œ , and sin U œ . Figure 5.48 shows 135° lies in quadrant II. The reference angle is
y
u¿ = 180° - 135° = 45°. 135 45 x
The function value for the reference angle is sin 45° =
22 . 2
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the function value in step 1. The angle u = 135° lies in quadrant II. Because the sine is positive in quadrant II, we put a + sign before the function value of the reference angle. Thus,
Figure 5.48 Reference angle for 135°
The sine is positive in quadrant II.
sin 135=±sin 45=
兹2 . 2
The reference angle for 135° is 45°.
b. We use our two-step procedure to find cos y 4p 3
x p 3
4p . 3
Step 1 Find the reference angle, U œ , and cos U œ . Figure 5.49 shows that 4p u = lies in quadrant III. The reference angle is 3 4p 3p p 4p - p = = . u¿ = 3 3 3 3 The function value for the reference angle is
Figure 5.49 Reference
cos
4p angle for 3
p 1 = . 3 2
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the 4p function value in step 1. The angle u = lies in quadrant III. Because only the 3 tangent and cotangent are positive in quadrant III, the cosine is negative in this quadrant. We put a - sign before the function value of the reference angle. Thus, The cosine is negative in quadrant III.
cos
1 p 4p =–cos =– . 2 3 3 The reference angle for 4p is p. 3 3
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522 Chapter 5 Trigonometric Functions c. We use our two-step procedure to find cot a -
Step 1 Find the reference angle, U œ , and cot U œ . Figure 5.50 shows that p p u = - lies in quadrant IV. The reference angle is u¿ = . The function value 3 3 p 23 for the reference angle is cot = . 3 3
y
x
p 3
p 3
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the p function value in step 1. The angle u = - lies in quadrant IV. Because only the 3 cosine and secant are positive in quadrant IV, the cotangent is negative in this quadrant. We put a - sign before the function value of the reference angle. Thus,
Figure 5.50 Reference angle for -
p b. 3
p 3
The cotangent is negative in quadrant IV.
p p 兹3 cot a– b =–cot =– . 3 3 3 The reference angle for − p is p. 3 3
7
Check Point Use reference angles to find the exact value of the following trigonometric functions: a. sin 300°
b. tan
c. sec a -
5p 4
p b. 6
In our final example, we use positive coterminal angles less than 2p to find the reference angles.
EXAMPLE 8
Using Reference Angles to Evaluate Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric functions: a. tan
14p 3
b. sec a -
17p b. 4
Solution 14p . 3 14p Step 1 Find the reference angle, U œ , and tan U œ . Because the given angle, 3 2 or 4 p, exceeds 2p, subtract 4p to find a positive coterminal angle less than 2p. 3 14p 14p 12p 2p u = - 4p = = 3 3 3 3
a. We use our two-step procedure to find tan
y
p 3
2p 3
x
Figure 5.51 Reference angle for
2p 3
2p Figure 5.51 shows u = in standard position. The angle lies in quadrant II. 3 The reference angle is 2p 3p 2p p u¿ = p = = . 3 3 3 3 p = 23. The function value for the reference angle is tan 3
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Section 5.3 Trigonometric Functions of Any Angle
523
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the 2p function value in step 1. The coterminal angle u = lies in quadrant II. 3 Because the tangent is negative in quadrant II, we put a - sign before the function value of the reference angle. Thus, The tangent is negative in quadrant II.
tan
p 2p 14p =tan =–tan =–兹3 . 3 3 3 The reference angle for 2p is p. 3 3
b. We use our two-step procedure to find sec a -
17p b. 4
Step 1 Find the reference angle, U œ , and sec U œ . Because the given angle, 17p 1 or -4 p, is less than -2p, add 6p (three multiples of 2p) to find a 4 4 positive coterminal angle less than 2p. u = -
Figure 5.52 shows u =
y
17p 17p 24p 7p + 6p = + = 4 4 4 4 7p in standard position. The angle lies in quadrant IV. 4
The reference angle is 7p 4 p 4
Figure 5.52 Reference angle for
7p 4
x
u¿ = 2p -
8p 7p p 7p = = . 4 4 4 4
The function value for the reference angle is sec
p = 22. 4
Step 2 Use the quadrant in which U lies to prefix the appropriate sign to the 7p function value in step 1. The coterminal angle u = lies in quadrant IV. 4 Because the secant is positive in quadrant IV, we put a + sign before the function value of the reference angle. Thus, The secant is positive in quadrant IV.
sec a–
17p 7p p b =sec =± sec =兹2 . 4 4 4 The reference angle for 7p is p. 4 4
Check Point
8
Use reference angles to find the exact value of each of the following trigonometric functions:
a. cos
17p 6
b. sin a -
22p b. 3
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524 Chapter 5 Trigonometric Functions Study Tip Evaluating trigonometric functions like those in Example 8 and Check Point 8 involves using a number of concepts, including finding coterminal angles and reference angles, locating special angles, determining the signs of trigonometric functions in specific quadrants, p p p and finding the trigonometric functions of special angles a 30° = , 45° = , and 60° = b. To be successful in trigonometry, it is 6 4 3 often necessary to connect concepts. Here’s an early reference sheet showing some of the concepts you should have at your fingertips (or memorized).
Degree and Radian Measures of Special and Quadrantal Angles p
90 , 2
2p
120 , 3 3p 135 , 4 5p
150 , 6
7p
11p
7p
330 , 6 7p 315 , 4 4p
240 , 3
3p
270 , 2
3p
270 , 2
210 , 6
0 , 0
210 , 6 5p 225 , 4
5p
300 , 3
180 , p
45
30°
U
1
30
P 6
5p
150 , 6 3p 135 , 4 2p 120 , 3
P 4
45°
p
30 , 6 p 45 , 4 p 60 , p 3
90 , 2
Signs of the Trigonometric Functions
60°
sin U
1 2
22 2
23 2
cos U
23 2
22 2
1 2
tan U
23 3
1
23
兹3
60
5p
300 , 3 7p 315 , 4 11p 330 , 6 0 , 0
Special Right Triangles and Trigonometric Functions of Special Angles
1
2
p
30 , 6
180 , p
兹2
4p
240 , 3 5p 225 , 4
p
60 , 3 p 45 , 4
P 3
y Quadrant II sine and cosecant positive
Quadrant I All functions positive x
Quadrant III tangent and cotangent positive
Quadrant IV cosine and secant positive
1
Using Reference Angles to Evaluate Trigonometric Functions
Trigonometric Functions of Quadrantal Angles U sin U cos U tan U
0° 0 0 1 0
90°
P 2
1 0 undefined
180° P 0 -1 0
270°
3P 2
-1 0 undefined
sin u= cos u= tan u=
sin u cos u tan u
+ or − in determined by the quadrant in which u lies and the sign of the function in that quadrant.
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525
Exercise Set 5.3 Practice Exercises In Exercises 1–8, a point on the terminal side of angle u is given. Find the exact value of each of the six trigonometric functions of u. 1. 1 -4, 32
2. 1- 12, 52
4. (3, 7)
7. 1- 2, -52
3. (2, 3)
5. 13, - 32
6. 15, -52
8. 1 - 1, -32
In Exercises 9–16, evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined. 9. cos p
10. tan p 3p 13. tan 2 p 16. tan 2
12. csc p p 15. cot 2
11. sec p 3p 14. cos 2
In Exercises 17–22, let u be an angle in standard position. Name the quadrant in which u lies. 17. sin u 7 0, cos u 7 0
18. sin u 6 0, cos u 7 0
19. sin u 6 0, cos u 6 0
20. tan u 6 0, sin u 6 0
21. tan u 6 0, cos u 6 0
22. cot u 7 0, sec u 6 0
In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of u. 23. cos u = - 35 ,
u in quadrant III
24. sin u = - 12 13 ,
u in quadrant III
25. sin u = 26. cos u = 27. cos u = 28. cos u = 29. tan u = 31. tan u =
5 13 , u in quadrant II 4 5 , u in quadrant IV 8 17 , 270° 6 u 6 360° 1 3 , 270° 6 u 6 360° - 23 , sin u 7 0 30. 4 32. 3 , cos u 6 0
33. sec u = - 3, tan u 7 0
tan u = - 13 , tan u =
5 12 ,
2p 3 7p 70. cot 4
7p 6 9p 72. tan 2
3p 4 9p 71. tan 4
67. sin
68. cos
69. csc
73. sin1 - 240°2
74. sin1 -225°2
75. tan a -
76. tan a -
77. sec 495°
78. sec 510°
p b 6 19p 79. cot 6 35p 82. cos 6 17p b 85. sin a 3
13p 3 17p b 83. tan a 6 35p b 86. sin a 6 80. cot
p b 4
23p 4 11p b 84. tan a 4 81. cos
Practice Plus In Exercises 87–92, find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. p p 3p cos p - cos sin 3 3 2 p p 88. sin cos 0 - sin cos p 4 6 11p 5p 11p cos + cos sin 89. sin 4 6 4 17p 5p 17p cos + cos sin 90. sin 3 4 3 87. sin
5p 6 5p 4
91. sin
3p 15p 5p tan a b - cos a b 2 4 3
92. sin
3p 8p 5p tan a b + cos a b 2 3 6
sin u 7 0 cos u 6 0
In Exercises 93–98, let f1x2 = sin x, g1x2 = cos x, and h1x2 = 2x.
34. csc u = - 4, tan u 7 0
Find the exact value of each expression. Do not use a calculator. In Exercises 35–60, find the reference angle for each angle. 35. 160° 38. 210° 7p 41. 4 5p 44. 7 47. - 335° 50. 5.5 17p 53. 6 17p 56. 3 25p 59. 6
36. 170° 39. 355° 5p 42. 4
37. 205° 40. 351° 5p 43. 6
45. - 150°
46. -250°
48. -359° 51. 565° 11p 54. 4 11p 57. 4 13p 60. 3
49. 4.7 52. 553° 23p 55. 4 17p 58. 6
93. f a
4p p 4p p + b + fa b + fa b 3 6 3 6
94. g a
5p p 5p p + b + ga b + ga b 6 6 6 6
95. 1h ⴰ g2a
96. 1h ⴰ f2a
11p b 4
97. the average rate of change of f from x1 = 98. the average rate of change of g from x1 =
5p 3p to x2 = 4 2
3p to x2 = p 4
In Exercises 99–104, find two values of u, 0 … u 6 2p, that satisfy each equation. 99. sin u =
In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator.
17p b 3
22 2
61. cos 225°
62. sin 300°
63. tan 210°
22 2
64. sec 240°
65. tan 420°
66. tan 405°
103. tan u = - 23
101. sin u = -
100. cos u =
1 2
102. cos u = -
1 2
104. tan u = -
23 3
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526 Chapter 5 Trigonometric Functions
Writing in Mathematics 105. If you are given a point on the terminal side of angle u, explain how to find sin u. 106. Explain why tan 90° is undefined.
113. When I found the exact value of cos 14p 3 , I used a number of concepts, including coterminal angles, reference angles, finding the cosine of a special angle, and knowing the cosine’s sign in various quadrants.
107. If cos u 7 0 and tan u 6 0, explain how to find the quadrant in which u lies. 108. What is a reference angle? Give an example with your description. 109. Explain how reference angles are used to evaluate trigonometric functions. Give an example with your description.
Critical Thinking Exercises Make Sense? In Exercises 110–113, determine whether each statement makes sense or does not make sense, and explain your reasoning. 110. I’m working with a quadrantal angle u for which sin u is undefined. 111. This angle u is in a quadrant in which sin u 6 0 and csc u 7 0. 112. I am given that tan u = 35 , so I can conclude that y = 3 and x = 5.
Section
5.4
Objectives
Preview Exercises Exercises 114–116 will help you prepare for the material covered in the next section. 114. Graph: x2 + y2 = 1. Then locate the point A - 12 , graph.
B on the
115. Use your graph of x2 + y2 = 1 from Exercise 114 to determine the relation’s domain and range. 116. a. Find the exact value of sin
A p4 B , sin A - p4 B , sin A p3 B , and
sin A - B . Based on your results, can the sine function be an even function? Explain your answer. p 3
b. Find the exact value of cos cos A -
p 3
A p4 B , cos A - p4 B , cos A p3 B , and
B . Based on your results, can the cosine
function be an odd function? Explain your answer.
Trigonometric Functions of Real Numbers; Periodic Functions
Use a unit circle to define
13 2
C
ycles govern many aspects of life—heartbeats, sleep patterns, seasons, and tides all follow regular, predictable cycles. In this section, we will see why trigonometric functions are used to model phenomena that occur in cycles. To do this, we need to move beyond angles and consider trigonometric functions of real numbers.
trigonometric functions of real numbers. Recognize the domain and range of sine and cosine functions. Use even and odd trigonometric functions. Use periodic properties.
Use a unit circle to define trigonometric functions of real numbers.
Trigonometric Functions of Real Numbers
y
1
s t x (1, 0)
x2 + y2 = 1
Figure 5.53 Unit circle with a central angle measuring t radians
Thus far, we have considered trigonometric functions of angles measured in degrees or radians. To define trigonometric functions of real numbers, rather than angles, we use a unit circle. A unit circle is a circle of radius 1, with its center at the origin of a rectangular coordinate system. The equation of this unit circle is x2 + y2 = 1. Figure 5.53 shows a unit circle in which the central angle measures t radians. We can use the formula for the length of a circular arc, s = ru, to find the length of the intercepted arc. s=ru=1 ⴢ t=t The radius of a unit circle is 1.
The radian measure of the central angle is t.
