Sines, Cosines, and Tangents
x 0)
Lesson 4-2
Lesson
4-2
Vocabulary
Sines, Cosines, and Tangents
unit circle cosine, cos sine, sin circular function
The sine and cosine functions relate magnitudes of rotations to coordinates of points on the unit circle. BIG IDEA
trigonometric function tangent, tan
The Sine, Cosine, and Tangent Functions
Mental Math
The unit circle is the circle with center at the origin and radius 1, as shown at the right.
The following regular polygons are inscribed in a circle. Find the measure of the angle formed by two rays from the center of the circle that contain adjacent vertices of the polygon.
Consider the rotation of magnitude θ with center at the origin. We call this Rθ. Regardless of the value of θ, the image of (1, 0) under Rθ is on the unit circle. Call this point P. We associate two numbers with each value of θ. The cosine of θ (abbreviated cos θ) is the x-coordinate of P; the sine of θ (abbreviated sin θ) is the y-coordinate of P.
P = Rθ (1, 0) = (cos θ, sin θ) y P θ O
x (1, 0)
a. triangle
b. square
c. pentagon d. octagon
Definition of cos θ and sin θ For all real numbers θ, (cos θ, sin θ) is the image of the point (1, 0) under a rotation of magnitude θ about the origin. That is, (cos θ, sin θ) = Rθ (1, 0). Because their definitions are based on a circle, the sine and cosine functions are sometimes called circular functions of θ. They are also called trigonometric functions, from the Greek word meaning “triangle measure,” as you will see in the applications for triangles presented in Chapter 5. To find values of trigonometric functions when θ is a multiple π of __ or 90º, you can use the above definition and mentally rotate (1, 0). 2
Example 1 Evaluate cos π and sin π. Solution Because no degree sign is given, π is measured in radians. Think of Rπ(1, 0), the image of (1, 0) under a rotation of π. Rπ(1, 0) = (–1, 0). So, by definition, (cos π, sin π) = (–1, 0). Thus, cos π = –1 and sin π = 0. Check Use a calculator. Make sure it is in radian mode.
Sines, Cosines, and Tangents
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229
4/28/09 9:14:51 AM
Chapter 4
Because π radians = 180º, Example 1 shows that cos 180º = –1 and π sin 180º = 0. Cosines and sines of other multiples of __ or 90º are shown 2 on the unit circle below. y
(0, 1) = (cos 90˚, sin 90˚) = cos π2 , sin π2
(-1, 0) = (cos 180˚, sin 180˚) = (cos π, sin π)
x (1, 0) = (cos 0˚, sin 0˚) = (cos 0, sin 0) (0, -1) = (cos 270˚, sin 270˚) 3π = cos 3π 2 , sin 2
The third most common circular function is defined in terms of the sine and cosine functions. The tangent of θ (abbreviated tan θ) equals the ratio of sin θ to cos θ.
Definition of Tangent sin θ For all real numbers θ, provided cos θ ≠ 0, tan θ = _____ . cos θ
When cos θ does equal zero, which occurs at any odd multiple of 90º, tan θ is undefi ned.
GUIDED
Example 2 a. Evaluate tan π.
b.
Evaluate tan (–270º).
Solution
a. From Example 1, "#$ π = ? %&' $(& π = ? . *# +%& π = ? . b. "#$ ,–-./0) = ? , $(& ,–-./0) = ? , $# +%& ,–-./0) ($ ? . For any value of θ, you can approximate sin θ, cos θ, and tan θ to the nearest tenth with a good drawing.
Activity
y
MATERIALS compass, protractor, graph paper, and calculator
1
Step 1 Work with a partner. Draw a set of coordinate axes on graph paper. Let each square on your grid have side length 0.1 unit. With the origin as center, draw a circle of radius 1. Label the figure as at the right. Step 2 a. Use a protractor to mark the image of A = (1, 0) under a rotation of 50º. Label this point P1, as shown at the right. b. Use the grid to estimate the x- and y-coordinates of P1. c. Estimate the slope of OP ##$1. d. Use your calculator to find cos 50º, sin 50º, and tan 50º. Make sure your calculator is set to degree mode.
230
P1
-1
O
Ax 1
-1
Trigonometric Functions
SMP_SEFST_C04L02_229_234_FINAL.i230 230
4/30/09 3:18:37 PM
Lesson 4-2
Step 3 a. Use a protractor to mark the image of A under R155º. Label this point P 2. b. Use the grid to estimate the x- and y-coordinates of P 2. Estimate the slope of OP !!"2. c. With your calculator, find cos 155º, sin 155º, and tan 155º. Step 4 Repeat Step 3 if the rotation has magnitude –100º. Call the image P 3. Step 5 How is tan θ related to the slope of the line through the origin and Rθ(A)? You can find better approximations to other values of sin θ, cos θ, or tan θ using a calculator.
Example 3
y B = R(1, 0)
Suppose the tips of the arms of a starfish determine the vertices of a regular pentagon. The point A = (1, 0) is at the tip of one arm, and so is one vertex of a regular pentagon ABCDE inscribed in the unit circle, as shown at the right. Find the coordinates of B to the nearest thousandth.
C x A = (1, 0) D
Solution Since the full circle measures 2π around, the measure of arc
2π 2! 2! ___ ___ AB is ___ 2π (1, 0). Consequently, B = (cos 5 , sin 5 ). A 5 . So B = R ___ 5
E
calculator shows B ≈ (0.309, 0.951).
y
For a given value of θ, you can determine whether (-, +) (+, +) sin θ, cos θ, and tan θ are positive or negative P without using a calculator by using coordinate θ x geometry and the unit circle. The cosine is positive when Rθ(1, 0) is in the first or fourth quadrant. The sine is positive when the image is in the first or (+, -) (-,-) second quadrant. The tangent is positive when the sine and cosine have the same sign and negative P = (cos θ, sin θ) = R!(1, 0) when they have opposite signs. The following table summarizes this information for values of θ between 0 and 360º or 2π. θ (radians)
θ (degrees)
quadrant of Rθ(1, 0)
cos θ
sin θ
tan θ
π 0 < θ < __
0º < θ < 90º
first
+
+
+
90º < θ < 180º
second
–
+
–
180º < θ < 270º
third
–
–
+
270º < θ < 360º
fourth
+
–
–
2
π __ 2
Lesson 4-2
Lesson
4-2
Vocabulary
Sines, Cosines, and Tangents
unit circle cosine, cos sine, sin circular function
The sine and cosine functions relate magnitudes of rotations to coordinates of points on the unit circle. BIG IDEA
trigonometric function tangent, tan
The Sine, Cosine, and Tangent Functions
Mental Math
The unit circle is the circle with center at the origin and radius 1, as shown at the right.
The following regular polygons are inscribed in a circle. Find the measure of the angle formed by two rays from the center of the circle that contain adjacent vertices of the polygon.
Consider the rotation of magnitude θ with center at the origin. We call this Rθ. Regardless of the value of θ, the image of (1, 0) under Rθ is on the unit circle. Call this point P. We associate two numbers with each value of θ. The cosine of θ (abbreviated cos θ) is the x-coordinate of P; the sine of θ (abbreviated sin θ) is the y-coordinate of P.
P = Rθ (1, 0) = (cos θ, sin θ) y P θ O
x (1, 0)
a. triangle
b. square
c. pentagon d. octagon
Definition of cos θ and sin θ For all real numbers θ, (cos θ, sin θ) is the image of the point (1, 0) under a rotation of magnitude θ about the origin. That is, (cos θ, sin θ) = Rθ (1, 0). Because their definitions are based on a circle, the sine and cosine functions are sometimes called circular functions of θ. They are also called trigonometric functions, from the Greek word meaning “triangle measure,” as you will see in the applications for triangles presented in Chapter 5. To find values of trigonometric functions when θ is a multiple π of __ or 90º, you can use the above definition and mentally rotate (1, 0). 2
Example 1 Evaluate cos π and sin π. Solution Because no degree sign is given, π is measured in radians. Think of Rπ(1, 0), the image of (1, 0) under a rotation of π. Rπ(1, 0) = (–1, 0). So, by definition, (cos π, sin π) = (–1, 0). Thus, cos π = –1 and sin π = 0. Check Use a calculator. Make sure it is in radian mode.
Sines, Cosines, and Tangents
SMP_SEFST_C04L02_229_234_FINAL.i229 229
229
4/28/09 9:14:51 AM
Chapter 4
Because π radians = 180º, Example 1 shows that cos 180º = –1 and π sin 180º = 0. Cosines and sines of other multiples of __ or 90º are shown 2 on the unit circle below. y
(0, 1) = (cos 90˚, sin 90˚) = cos π2 , sin π2
(-1, 0) = (cos 180˚, sin 180˚) = (cos π, sin π)
x (1, 0) = (cos 0˚, sin 0˚) = (cos 0, sin 0) (0, -1) = (cos 270˚, sin 270˚) 3π = cos 3π 2 , sin 2
The third most common circular function is defined in terms of the sine and cosine functions. The tangent of θ (abbreviated tan θ) equals the ratio of sin θ to cos θ.
Definition of Tangent sin θ For all real numbers θ, provided cos θ ≠ 0, tan θ = _____ . cos θ
When cos θ does equal zero, which occurs at any odd multiple of 90º, tan θ is undefi ned.
GUIDED
Example 2 a. Evaluate tan π.
b.
Evaluate tan (–270º).
Solution
a. From Example 1, "#$ π = ? %&' $(& π = ? . *# +%& π = ? . b. "#$ ,–-./0) = ? , $(& ,–-./0) = ? , $# +%& ,–-./0) ($ ? . For any value of θ, you can approximate sin θ, cos θ, and tan θ to the nearest tenth with a good drawing.
Activity
y
MATERIALS compass, protractor, graph paper, and calculator
1
Step 1 Work with a partner. Draw a set of coordinate axes on graph paper. Let each square on your grid have side length 0.1 unit. With the origin as center, draw a circle of radius 1. Label the figure as at the right. Step 2 a. Use a protractor to mark the image of A = (1, 0) under a rotation of 50º. Label this point P1, as shown at the right. b. Use the grid to estimate the x- and y-coordinates of P1. c. Estimate the slope of OP ##$1. d. Use your calculator to find cos 50º, sin 50º, and tan 50º. Make sure your calculator is set to degree mode.
230
P1
-1
O
Ax 1
-1
Trigonometric Functions
SMP_SEFST_C04L02_229_234_FINAL.i230 230
4/30/09 3:18:37 PM
Lesson 4-2
Step 3 a. Use a protractor to mark the image of A under R155º. Label this point P 2. b. Use the grid to estimate the x- and y-coordinates of P 2. Estimate the slope of OP !!"2. c. With your calculator, find cos 155º, sin 155º, and tan 155º. Step 4 Repeat Step 3 if the rotation has magnitude –100º. Call the image P 3. Step 5 How is tan θ related to the slope of the line through the origin and Rθ(A)? You can find better approximations to other values of sin θ, cos θ, or tan θ using a calculator.
Example 3
y B = R(1, 0)
Suppose the tips of the arms of a starfish determine the vertices of a regular pentagon. The point A = (1, 0) is at the tip of one arm, and so is one vertex of a regular pentagon ABCDE inscribed in the unit circle, as shown at the right. Find the coordinates of B to the nearest thousandth.
C x A = (1, 0) D
Solution Since the full circle measures 2π around, the measure of arc
2π 2! 2! ___ ___ AB is ___ 2π (1, 0). Consequently, B = (cos 5 , sin 5 ). A 5 . So B = R ___ 5
E
calculator shows B ≈ (0.309, 0.951).
y
For a given value of θ, you can determine whether (-, +) (+, +) sin θ, cos θ, and tan θ are positive or negative P without using a calculator by using coordinate θ x geometry and the unit circle. The cosine is positive when Rθ(1, 0) is in the first or fourth quadrant. The sine is positive when the image is in the first or (+, -) (-,-) second quadrant. The tangent is positive when the sine and cosine have the same sign and negative P = (cos θ, sin θ) = R!(1, 0) when they have opposite signs. The following table summarizes this information for values of θ between 0 and 360º or 2π. θ (radians)
θ (degrees)
quadrant of Rθ(1, 0)
cos θ
sin θ
tan θ
π 0 < θ < __
0º < θ < 90º
first
+
+
+
90º < θ < 180º
second
–
+
–
180º < θ < 270º
third
–
–
+
270º < θ < 360º
fourth
+
–
–
2
π __ 2
