5.2 Trigonometric Ratios of Any Angle
5.2 Trigonometric Ratios of Any Angle The use of cranes to lift heavy objects is an essential part of the construction and shipping industries. There are many different designs of crane, but they usually include some kind of winding mechanism, called a winch, which pulls or releases a cable. In one crane design, the cable passes along a straight tube, called a boom or jib, which can be raised or lowered by changing its angle of inclination. The cable passes over a pulley at the upper end of the boom. When the hook at the end of the cable is attached to a heavy object, the winch is used to raise or lower it. Operating a large crane is a highly skilled occupation. In adjusting the position of the boom, the operator uses the concepts of trigonometry, as will be shown in Example 5. I NVESTIGATE & I NQUIRE Angles that measure 30°, 45°, and 60° occur often in trigonometry. They are sometimes called special angles. Right triangles in which they are found are sometimes called special triangles.
A 60° 2
2
ABC is an equilateral triangle with side lengths of 2 units. AD bisects ∠BAC to form two congruent triangles, ABD and ACD. ABD and ACD are called 30°-60°-90° B triangles. What is the length of BD?
1.
2.
D 2
C
Determine the length of AD. Express the answer in radical form.
3.
Repeat questions 1 and 2, beginning with equilateral triangles with the following side lengths. Express the length of AD as a mixed radical. a) 4 units b) 10 units Describe the relationship between the side lengths in any 30°-60°-90° triangle. 4.
5.2 Trigonometric Ratios of Any Angle • MHR 341
Use one of the 30°-60°-90° triangles to determine the exact values of the sine, cosine, and tangent of 30° and 60°. 5.
E
DEF is an isosceles right triangle whose equal sides, DE and DF, have a length of 1 unit. DEF is called a 45°-45°-90° triangle. Determine the length of EF. Express the answer in radical form.
6.
45° 1 45° F
1
D
7. Repeat question 6, beginning with isosceles right triangles whose equal sides have the following lengths. Express the length of EF as a mixed radical. a) 2 units b) 6 units
Describe the relationship between the lengths of the sides in any 45°-45°-90° triangle. 8.
Use one of the 45°-45°-90° triangles to determine the exact values of the sine, cosine, and tangent of 45°. 9.
The angle θ is shown in standard position. The point P(x, y) is any point on the terminal arm. 2 By the Pythagorean theorem, r = x. + y2 Recall that you have found the sine and cosine ratios of an angle θ in standard position, where 0° ≤ θ ≤ 180°. We will now extend the domain of θ to include angles of any measure, and will include the tangent ratio of any angle in standard position.
y P(x,y) r y
For any angle θ in standard position, with point P(x, y) on the terminal 2 arm, and r = x, + y2 the three primary trigonometric ratios are defined in terms of x, y, and r as follows. y y cos θ = x tan θ = sin θ = r x r EXAMPLE 1 Sine, Cosine, and Tangent of Any Angle The point P(−3, −6) lies on the terminal arm of an angle θ in standard position. Determine the exact values of sin θ, cos θ, and tan θ.
342 MHR • Chapter 5
x 0
θ
x
SOLUTION
y
Sketch the angle in standard position. Construct PAO by drawing a line segment perpendicular to the x-axis from the point P(−3, −6). r2 = x2 + y2 2 r = x + y2 2 = (–3) + (–6)2 = 9 + 36 = 45 = 35 y y x sin θ = cos θ = tan θ = r r x –6 –6 –3 = = = –3 35 35 =2 –2 or − 2 –1 or − 1 = = 5 5 5 5
θ
A –3
x
O
–6
r P(–3, –6)
2 , cos θ = – 1 , and tan θ = 2. So, sin θ = – 5 5 This chart summarizes the signs of trigonometric ratios. y Quadrant II Quadrant I y
P r
y
θ
x
x
y
θ
x y
r P
x
90° < θ < 180° y + sin θ = = + r + – x cos θ = = − + r y + tan θ = = − x –
0° < θ < 90° y sin θ = r cos θ = x r y tan θ = x
Quadrant III 180° < θ < 270° y – sin θ = = − r + x – cos θ = = − r + y –= + tan θ = x –
Quadrant IV 270° < θ < 360° y – sin θ = = − r + + cos θ = x = + r + y – tan θ = = − x +
y
+ = + + + = + + + = + +
P(x,y)
r
y
θ x
0
x
x
y θ
x y
x
r P
5.2 Trigonometric Ratios of Any Angle • MHR 343
The memory device CAST shows which trigonometric ratios are positive in each quadrant.
The Pythagorean theorem shows that the side lengths of a 30°-60°-90° triangle have a ratio of 1 to 3 to 2, or 1:3:2. The side lengths of a 45°-45°-90° triangle have a ratio of 1 to 1 to 2 , or 1:1:2. These special triangles are used to find the exact trigonometric ratios of special angles. θ in Degrees θ in Radians 30° 45° 60°
sin θ
cos θ
tan θ
π 6
1 2
3 2
π 4 π 3
1 3
1 2 3 2
1 2 1 2
1
y S sine
A all
T tangent
C x cosine
30°
45° 3
2
2
60°
1
45° 1
1
3
EXAMPLE 2 Exact Trigonometric Ratios from Degree Measures Find the exact values of the sine, cosine, and tangent of 120°. SOLUTION Sketch the angle in standard position. Construct a triangle by drawing a perpendicular from the terminal arm to the x-axis. 3 The angle between the terminal arm and the x-axis is 60°. So, the triangle formed is a 30°-60°-90° triangle. Choose point P on the terminal arm so that r = 2. It follows that x = −1 and y = 3. y y cos θ = x tan θ = sin θ = r x r –1 or − 1 3 3 or −3 sin 120° = cos 120° = tan 120° = 2 2 –1 2 3 , cos 120° = – 1 , and tan 120° = –3. So, sin 120° = 2 2
344 MHR • Chapter 5
P
y 2 120°
–1
x
EXAMPLE 3 Exact Trigonometric Ratios from Radian Measures Find the exact values of 7 5 a) sin π b) cos π 4 6 SOLUTION y
Sketch the angle in standard position. Construct a triangle by drawing a perpendicular from the terminal arm to the x-axis. 7 7 180 ° π rad = π 4 4 π = 315° The angle between the terminal arm and the x-axis is 45°. So, the triangle formed is a 45°-45°-90° triangle. Choose point P on the terminal arm so that r = 2. It follows that x = 1 and y = −1. y sin θ = r –1 or − 1 sin 7π = 4 2 2 7π = – 1 . So, sin 4 2 a)
Sketch the angle in standard position. Construct a triangle by drawing a perpendicular from the terminal arm to the x-axis. 5 5 180 ° π rad = π 6 6 π = 150°
1 7 _π 4
2 P
b)
x –1
y P 1
2 – 3
5 _π 6
x
The angle between the terminal arm and the x-axis is 30°. So, the triangle formed is a 30°-60°-90° triangle. Choose point P on the terminal arm so that r = 2. It follows that x = –3 and y = 1. cos θ = x r –3 or − 3 cos 5π = 6 2 2 3 . So, cos 5π = – 6 2 5.2 Trigonometric Ratios of Any Angle • MHR 345
EXAMPLE 4 Trigonometric Ratios of a Right Angle Find the values of the sine, cosine, and tangent of an angle that measures 90°. SOLUTION Sketch the angle in standard position. Choose P(0, 1) on the terminal arm of the angle. Therefore, x = 0, y = 1, and r = 1. y y sin θ = cos θ = x tan θ = r x r 1 0 1 sin 90° = cos 90° = tan 90° = 1 1 0 =1 =0 Since division by 0 is not defined, tan 90° is not defined.
y P(0,1) 90°
x
0
So, sin 90° = 1, cos 90° = 0, and tan 90° is not defined. EXAMPLE 5 Operating a Crane The boom of a crane can be moved so that its angle of inclination changes. In one location, close to some buildings and some overhead power cables, the minimum value of the angle of inclination is 30°, and the maximum value is 60°. If the boom is 10 m long, find the vertical displacement of the end of the boom when the angle of inclination increases from its minimum value to its maximum value. Express your answer in exact form and in approximate form, to the nearest tenth of a metre. SOLUTION a) In the diagram, the vertical displacement is y2 − y1. Using the exact values of the trigonometric ratios for 30° and 60°, write expressions for y2 and y1. y1 y2 sin 30° = sin 60° = 10 10 1 y1 3 y2 = = 2 2 10 10 10 103 = y1 = y2 2 2 5 = y1 53 = y2 346 MHR • Chapter 5
vertical displacement
y 10 m
60° 30°
y2 y1
x
The vertical displacement is y2 − y1 = 53 − 5 =⋅ 3.7
The vertical displacement is 53 − 5 m in exact form, or 3.7 m in approximate form, to the nearest tenth of a metre. Key
Concepts
• For any angle θ in standard position, with point P(x, y) on the terminal arm, 2 + y2 the three primary trigonometric ratios are defined in terms and r = x, of x, y, and r as follows. y y sin θ = cos θ = x tan θ = r x r y • The memory device CAST shows which trigonometric A S ratios are positive in each quadrant. all sine T tangent
• The two special triangles used to find the exact values of the sine, cosine, and tangent of special angles are as shown.
30°-60°-90° triangle
45° 2
3
1
45° 1
Yo u r
45°-45°-90° triangle
30° 2
60°
Communicate
C x cosine
1
Understanding
The point P(3, −4) lies on the terminal arm of an angle θ in standard position. Describe how you would determine the exact values of sin θ, cos θ, and tan θ. 2 2. Describe how you would find the exact value of tan π. 3 3. Describe how you would find the exact value of sin 135°. 4. Describe how you would find the exact value of cos 270°. 1.
5.2 Trigonometric Ratios of Any Angle • MHR 347
Practise A The coordinates of a point P on the terminal arm of each ∠θ in standard position are shown, where 0 ≤ θ ≤ 2π. Determine the exact values of sin θ, cos θ, and tan θ. 1.
a)
b)
y
y P(–3, 5)
P(8,15)
θ
θ
0
x
0
y
c)
0 x
y θ
x
0 P(12, –5)
P(–4, –3)
e)
θ
y 0
Find the exact value of each trigonometric ratio. a) sin 30° b) tan 315° c) cos 240° d) tan 150° e) cos 225° f) sin 45° g) cos 330° h) sin 300° 3.
d)
θ
x
The coordinates of a point P on a terminal arm of an ∠θ in standard position are given, where 0 ≤ θ ≤ 2π. Determine the exact values of sin θ, cos θ, and tan θ. a) P(6, 5) b) P(−1, 8) c) P(−2, −5) d) P(6, −1) e) P(2, −4) f) P(−3, −9) g) P(3, 3) h) P(−2, 6) 2.
y
f) x θ
Find the exact value of each trigonometric ratio. 5 11 a) sin π b) tan π 4 6 π 7 c) cos d) cos π 6 4 4 7 e) tan π f) cos π 3 6 5 3 g) sin π h) cos π 6 4 4.
x
0 P(3, –2)
P(–2, –7)
Apply, Solve, Communicate The arm of a boom crane is 12 m long. Because of the location of the construction site, the angle of inclination of the boom of the crane has a minimum value of 30° and a maximum value of 45°. Find the vertical displacement of the end of the boom as a) an exact value b) an approximate value, to the nearest tenth of a metre 5. Boom crane
348 MHR • Chapter 5
B ∠θ is in standard position with its terminal arm in the stated quadrant, and 0 ≤ θ ≤ 2π. A trigonometric ratio is given. Find the exact values of the other two trigonometric ratios. 4 2 a) sin θ = , quadrant II b) cos θ = – , quadrant III 5 3 5 3 c) tan θ = – , quadrant IV d) sin θ = – , quadrant III 2 7 6.
∠θ is in standard position, and 0 ≤ θ ≤ 2π. A trigonometric ratio is given. Find the exact values of the other two trigonometric ratios. 1 3 1 a) sin θ = b) cos θ = c) tan θ = 3 5 4 1 8 5 d) cos θ = – e) tan θ = – f) sin θ = – 2 5 6 7.
A cargo ship is tied up at a dock. At low tide, a 12-m long unloading ramp slopes down from the ship to the dock and makes an angle of 30° to the horizontal. At high tide, the ship is closer to the dock, and the unloading ramp makes an angle of 45° to the horizontal. a) Determine the change in the horizontal distance from the ship to the dock from low tide to high tide. Express the distance as an exact value and as an approximate value, to the nearest hundredth of a metre. b) Determine the change in the height of the upper end of the unloading ramp above the dock from low tide to high tide. Express the change in height as an exact value and as an approximate value, to the nearest hundredth of a metre. 8. Application
Bridges that are hinged so that they can be raised are known as bascule bridges. They are commonly built over water, where they can be opened to 80 m allow large boats to pass through. Some bascule bridges are hinged at one end, but others are built in two halves that separate when they are raised. A bascule bridge built in two halves is 80 m long. When the bridge is opened fully, each half of the bridge makes an angle of 60° with the horizontal. When the bridge is opened fully, how far apart are the top ends of the two halves of the bridge?
9. Bascule bridges
A flagpole is anchored by two pairs of guy wires attached 8 m above the ground. One pair is anchored to the ground at 45° angles. The other pair is anchored to the ground at 60° angles. Which pair of guy wires has the greater total length and by how much, to the nearest 45° tenth of a metre?
60°
60° 80 m
10. Flagpole
60°
60°
45°
5.2 Trigonometric Ratios of Any Angle • MHR 349
If 0° ≤ A ≤ 360°, find the possible measures of ∠A. 1 3 1 a) sin A = b) cos A = c) tan A = 1 d) sin A = 2 2 2 3 h) sin A = – 1 1 e) cos A = – f) tan A = –3 g) cos A = – 2 2 2 1 1 i) tan A = –1 j) tan A = – k) cos A = – 2 3 11.
table shows the trigonometric ratios of a right angle, from Example 4. Copy the sin θ table and complete it by finding the cos θ trigonometric ratios of 0°, 180°, 270°, and tan θ 360° angles. 1 13. If 0 ≤ A ≤ 2π, and cos A = , find the values of 2 a) sin A b) tan A Ratio
12. The
14. If 0 ≤ a) sin A
Angle Measure 0°
90° 1
180°
0 n.d.
A ≤ 2π, and tan A = –3, find the values of b) cos A
C If the shortest side of a 30°-60°-90° triangle is equal in length to the shortest side of a 45°-45°-90° triangle, what is the ratio of the perimeter of the first triangle to the perimeter of the second triangle, to the nearest hundredth? 15. Inquiry/Problem Solving
If the longest side of a 30°-60°-90° triangle is equal in length to the shortest side of a 45°-45°-90° triangle, what is the ratio of the area of the first triangle to the area of the second triangle, to the nearest hundredth?
16.
The terminal arm of angle θ in standard position has a slope of −1. If 0 ≤ θ ≤ 2π, what are the possible values of the following? Explain. a) sin θ b) cos θ c) tan θ 17. Communication
18. Find the exact value of each of a) sin 30°sin 45°sin 60° c) sin 60°cos 30° + sin 30°cos 60° e)
5 tan 60° cos 30°
350 MHR • Chapter 5
the following. b) sin 30°cos 30° + sin 60°cos 60° d) 2sin 30°cos 30°
270°
360°
A 60° 2
2
ABC is an equilateral triangle with side lengths of 2 units. AD bisects ∠BAC to form two congruent triangles, ABD and ACD. ABD and ACD are called 30°-60°-90° B triangles. What is the length of BD?
1.
2.
D 2
C
Determine the length of AD. Express the answer in radical form.
3.
Repeat questions 1 and 2, beginning with equilateral triangles with the following side lengths. Express the length of AD as a mixed radical. a) 4 units b) 10 units Describe the relationship between the side lengths in any 30°-60°-90° triangle. 4.
5.2 Trigonometric Ratios of Any Angle • MHR 341
Use one of the 30°-60°-90° triangles to determine the exact values of the sine, cosine, and tangent of 30° and 60°. 5.
E
DEF is an isosceles right triangle whose equal sides, DE and DF, have a length of 1 unit. DEF is called a 45°-45°-90° triangle. Determine the length of EF. Express the answer in radical form.
6.
45° 1 45° F
1
D
7. Repeat question 6, beginning with isosceles right triangles whose equal sides have the following lengths. Express the length of EF as a mixed radical. a) 2 units b) 6 units
Describe the relationship between the lengths of the sides in any 45°-45°-90° triangle. 8.
Use one of the 45°-45°-90° triangles to determine the exact values of the sine, cosine, and tangent of 45°. 9.
The angle θ is shown in standard position. The point P(x, y) is any point on the terminal arm. 2 By the Pythagorean theorem, r = x. + y2 Recall that you have found the sine and cosine ratios of an angle θ in standard position, where 0° ≤ θ ≤ 180°. We will now extend the domain of θ to include angles of any measure, and will include the tangent ratio of any angle in standard position.
y P(x,y) r y
For any angle θ in standard position, with point P(x, y) on the terminal 2 arm, and r = x, + y2 the three primary trigonometric ratios are defined in terms of x, y, and r as follows. y y cos θ = x tan θ = sin θ = r x r EXAMPLE 1 Sine, Cosine, and Tangent of Any Angle The point P(−3, −6) lies on the terminal arm of an angle θ in standard position. Determine the exact values of sin θ, cos θ, and tan θ.
342 MHR • Chapter 5
x 0
θ
x
SOLUTION
y
Sketch the angle in standard position. Construct PAO by drawing a line segment perpendicular to the x-axis from the point P(−3, −6). r2 = x2 + y2 2 r = x + y2 2 = (–3) + (–6)2 = 9 + 36 = 45 = 35 y y x sin θ = cos θ = tan θ = r r x –6 –6 –3 = = = –3 35 35 =2 –2 or − 2 –1 or − 1 = = 5 5 5 5
θ
A –3
x
O
–6
r P(–3, –6)
2 , cos θ = – 1 , and tan θ = 2. So, sin θ = – 5 5 This chart summarizes the signs of trigonometric ratios. y Quadrant II Quadrant I y
P r
y
θ
x
x
y
θ
x y
r P
x
90° < θ < 180° y + sin θ = = + r + – x cos θ = = − + r y + tan θ = = − x –
0° < θ < 90° y sin θ = r cos θ = x r y tan θ = x
Quadrant III 180° < θ < 270° y – sin θ = = − r + x – cos θ = = − r + y –= + tan θ = x –
Quadrant IV 270° < θ < 360° y – sin θ = = − r + + cos θ = x = + r + y – tan θ = = − x +
y
+ = + + + = + + + = + +
P(x,y)
r
y
θ x
0
x
x
y θ
x y
x
r P
5.2 Trigonometric Ratios of Any Angle • MHR 343
The memory device CAST shows which trigonometric ratios are positive in each quadrant.
The Pythagorean theorem shows that the side lengths of a 30°-60°-90° triangle have a ratio of 1 to 3 to 2, or 1:3:2. The side lengths of a 45°-45°-90° triangle have a ratio of 1 to 1 to 2 , or 1:1:2. These special triangles are used to find the exact trigonometric ratios of special angles. θ in Degrees θ in Radians 30° 45° 60°
sin θ
cos θ
tan θ
π 6
1 2
3 2
π 4 π 3
1 3
1 2 3 2
1 2 1 2
1
y S sine
A all
T tangent
C x cosine
30°
45° 3
2
2
60°
1
45° 1
1
3
EXAMPLE 2 Exact Trigonometric Ratios from Degree Measures Find the exact values of the sine, cosine, and tangent of 120°. SOLUTION Sketch the angle in standard position. Construct a triangle by drawing a perpendicular from the terminal arm to the x-axis. 3 The angle between the terminal arm and the x-axis is 60°. So, the triangle formed is a 30°-60°-90° triangle. Choose point P on the terminal arm so that r = 2. It follows that x = −1 and y = 3. y y cos θ = x tan θ = sin θ = r x r –1 or − 1 3 3 or −3 sin 120° = cos 120° = tan 120° = 2 2 –1 2 3 , cos 120° = – 1 , and tan 120° = –3. So, sin 120° = 2 2
344 MHR • Chapter 5
P
y 2 120°
–1
x
EXAMPLE 3 Exact Trigonometric Ratios from Radian Measures Find the exact values of 7 5 a) sin π b) cos π 4 6 SOLUTION y
Sketch the angle in standard position. Construct a triangle by drawing a perpendicular from the terminal arm to the x-axis. 7 7 180 ° π rad = π 4 4 π = 315° The angle between the terminal arm and the x-axis is 45°. So, the triangle formed is a 45°-45°-90° triangle. Choose point P on the terminal arm so that r = 2. It follows that x = 1 and y = −1. y sin θ = r –1 or − 1 sin 7π = 4 2 2 7π = – 1 . So, sin 4 2 a)
Sketch the angle in standard position. Construct a triangle by drawing a perpendicular from the terminal arm to the x-axis. 5 5 180 ° π rad = π 6 6 π = 150°
1 7 _π 4
2 P
b)
x –1
y P 1
2 – 3
5 _π 6
x
The angle between the terminal arm and the x-axis is 30°. So, the triangle formed is a 30°-60°-90° triangle. Choose point P on the terminal arm so that r = 2. It follows that x = –3 and y = 1. cos θ = x r –3 or − 3 cos 5π = 6 2 2 3 . So, cos 5π = – 6 2 5.2 Trigonometric Ratios of Any Angle • MHR 345
EXAMPLE 4 Trigonometric Ratios of a Right Angle Find the values of the sine, cosine, and tangent of an angle that measures 90°. SOLUTION Sketch the angle in standard position. Choose P(0, 1) on the terminal arm of the angle. Therefore, x = 0, y = 1, and r = 1. y y sin θ = cos θ = x tan θ = r x r 1 0 1 sin 90° = cos 90° = tan 90° = 1 1 0 =1 =0 Since division by 0 is not defined, tan 90° is not defined.
y P(0,1) 90°
x
0
So, sin 90° = 1, cos 90° = 0, and tan 90° is not defined. EXAMPLE 5 Operating a Crane The boom of a crane can be moved so that its angle of inclination changes. In one location, close to some buildings and some overhead power cables, the minimum value of the angle of inclination is 30°, and the maximum value is 60°. If the boom is 10 m long, find the vertical displacement of the end of the boom when the angle of inclination increases from its minimum value to its maximum value. Express your answer in exact form and in approximate form, to the nearest tenth of a metre. SOLUTION a) In the diagram, the vertical displacement is y2 − y1. Using the exact values of the trigonometric ratios for 30° and 60°, write expressions for y2 and y1. y1 y2 sin 30° = sin 60° = 10 10 1 y1 3 y2 = = 2 2 10 10 10 103 = y1 = y2 2 2 5 = y1 53 = y2 346 MHR • Chapter 5
vertical displacement
y 10 m
60° 30°
y2 y1
x
The vertical displacement is y2 − y1 = 53 − 5 =⋅ 3.7
The vertical displacement is 53 − 5 m in exact form, or 3.7 m in approximate form, to the nearest tenth of a metre. Key
Concepts
• For any angle θ in standard position, with point P(x, y) on the terminal arm, 2 + y2 the three primary trigonometric ratios are defined in terms and r = x, of x, y, and r as follows. y y sin θ = cos θ = x tan θ = r x r y • The memory device CAST shows which trigonometric A S ratios are positive in each quadrant. all sine T tangent
• The two special triangles used to find the exact values of the sine, cosine, and tangent of special angles are as shown.
30°-60°-90° triangle
45° 2
3
1
45° 1
Yo u r
45°-45°-90° triangle
30° 2
60°
Communicate
C x cosine
1
Understanding
The point P(3, −4) lies on the terminal arm of an angle θ in standard position. Describe how you would determine the exact values of sin θ, cos θ, and tan θ. 2 2. Describe how you would find the exact value of tan π. 3 3. Describe how you would find the exact value of sin 135°. 4. Describe how you would find the exact value of cos 270°. 1.
5.2 Trigonometric Ratios of Any Angle • MHR 347
Practise A The coordinates of a point P on the terminal arm of each ∠θ in standard position are shown, where 0 ≤ θ ≤ 2π. Determine the exact values of sin θ, cos θ, and tan θ. 1.
a)
b)
y
y P(–3, 5)
P(8,15)
θ
θ
0
x
0
y
c)
0 x
y θ
x
0 P(12, –5)
P(–4, –3)
e)
θ
y 0
Find the exact value of each trigonometric ratio. a) sin 30° b) tan 315° c) cos 240° d) tan 150° e) cos 225° f) sin 45° g) cos 330° h) sin 300° 3.
d)
θ
x
The coordinates of a point P on a terminal arm of an ∠θ in standard position are given, where 0 ≤ θ ≤ 2π. Determine the exact values of sin θ, cos θ, and tan θ. a) P(6, 5) b) P(−1, 8) c) P(−2, −5) d) P(6, −1) e) P(2, −4) f) P(−3, −9) g) P(3, 3) h) P(−2, 6) 2.
y
f) x θ
Find the exact value of each trigonometric ratio. 5 11 a) sin π b) tan π 4 6 π 7 c) cos d) cos π 6 4 4 7 e) tan π f) cos π 3 6 5 3 g) sin π h) cos π 6 4 4.
x
0 P(3, –2)
P(–2, –7)
Apply, Solve, Communicate The arm of a boom crane is 12 m long. Because of the location of the construction site, the angle of inclination of the boom of the crane has a minimum value of 30° and a maximum value of 45°. Find the vertical displacement of the end of the boom as a) an exact value b) an approximate value, to the nearest tenth of a metre 5. Boom crane
348 MHR • Chapter 5
B ∠θ is in standard position with its terminal arm in the stated quadrant, and 0 ≤ θ ≤ 2π. A trigonometric ratio is given. Find the exact values of the other two trigonometric ratios. 4 2 a) sin θ = , quadrant II b) cos θ = – , quadrant III 5 3 5 3 c) tan θ = – , quadrant IV d) sin θ = – , quadrant III 2 7 6.
∠θ is in standard position, and 0 ≤ θ ≤ 2π. A trigonometric ratio is given. Find the exact values of the other two trigonometric ratios. 1 3 1 a) sin θ = b) cos θ = c) tan θ = 3 5 4 1 8 5 d) cos θ = – e) tan θ = – f) sin θ = – 2 5 6 7.
A cargo ship is tied up at a dock. At low tide, a 12-m long unloading ramp slopes down from the ship to the dock and makes an angle of 30° to the horizontal. At high tide, the ship is closer to the dock, and the unloading ramp makes an angle of 45° to the horizontal. a) Determine the change in the horizontal distance from the ship to the dock from low tide to high tide. Express the distance as an exact value and as an approximate value, to the nearest hundredth of a metre. b) Determine the change in the height of the upper end of the unloading ramp above the dock from low tide to high tide. Express the change in height as an exact value and as an approximate value, to the nearest hundredth of a metre. 8. Application
Bridges that are hinged so that they can be raised are known as bascule bridges. They are commonly built over water, where they can be opened to 80 m allow large boats to pass through. Some bascule bridges are hinged at one end, but others are built in two halves that separate when they are raised. A bascule bridge built in two halves is 80 m long. When the bridge is opened fully, each half of the bridge makes an angle of 60° with the horizontal. When the bridge is opened fully, how far apart are the top ends of the two halves of the bridge?
9. Bascule bridges
A flagpole is anchored by two pairs of guy wires attached 8 m above the ground. One pair is anchored to the ground at 45° angles. The other pair is anchored to the ground at 60° angles. Which pair of guy wires has the greater total length and by how much, to the nearest 45° tenth of a metre?
60°
60° 80 m
10. Flagpole
60°
60°
45°
5.2 Trigonometric Ratios of Any Angle • MHR 349
If 0° ≤ A ≤ 360°, find the possible measures of ∠A. 1 3 1 a) sin A = b) cos A = c) tan A = 1 d) sin A = 2 2 2 3 h) sin A = – 1 1 e) cos A = – f) tan A = –3 g) cos A = – 2 2 2 1 1 i) tan A = –1 j) tan A = – k) cos A = – 2 3 11.
table shows the trigonometric ratios of a right angle, from Example 4. Copy the sin θ table and complete it by finding the cos θ trigonometric ratios of 0°, 180°, 270°, and tan θ 360° angles. 1 13. If 0 ≤ A ≤ 2π, and cos A = , find the values of 2 a) sin A b) tan A Ratio
12. The
14. If 0 ≤ a) sin A
Angle Measure 0°
90° 1
180°
0 n.d.
A ≤ 2π, and tan A = –3, find the values of b) cos A
C If the shortest side of a 30°-60°-90° triangle is equal in length to the shortest side of a 45°-45°-90° triangle, what is the ratio of the perimeter of the first triangle to the perimeter of the second triangle, to the nearest hundredth? 15. Inquiry/Problem Solving
If the longest side of a 30°-60°-90° triangle is equal in length to the shortest side of a 45°-45°-90° triangle, what is the ratio of the area of the first triangle to the area of the second triangle, to the nearest hundredth?
16.
The terminal arm of angle θ in standard position has a slope of −1. If 0 ≤ θ ≤ 2π, what are the possible values of the following? Explain. a) sin θ b) cos θ c) tan θ 17. Communication
18. Find the exact value of each of a) sin 30°sin 45°sin 60° c) sin 60°cos 30° + sin 30°cos 60° e)
5 tan 60° cos 30°
350 MHR • Chapter 5
the following. b) sin 30°cos 30° + sin 60°cos 60° d) 2sin 30°cos 30°
270°
360°
