Ch. 1 - Angles and the Trigonometric Functions
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- In Exercises 28–29, find a cofunction with the same value as the given expression. sin 70°
Problem 28
- In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = −4 sin 3π/2 t
Problem 28
- In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sin² 𝜋 + cos² 𝜋 10 10
Problem 28
- In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sec² 23° - tan² 23°
Problem 29
- In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
Problem 29
tan 𝜋
- In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
Problem 30
cot 15𝜋/2
Problem 30a
In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of
0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋.
4 2 4 4 2 4
a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function.
b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number.
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cot 𝜋/2
- In Exercises 31–38, find a cofunction with the same value as the given expression. sin 7°
Problem 31
- In Exercises 25–32, the unit circle has been divided into eight equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 3𝜋, 𝜋, 5𝜋, 3𝜋, 7𝜋, and 2𝜋. 4 2 4 4 2 4 a. Use the (x,y) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. sin 47𝜋/4
Problem 31
Problem 32
Find a cofunction with the same value as the given expression.
sin 19°
- In Exercises 31–38, find a cofunction with the same value as the given expression. csc 25°
Problem 33
Problem 34
Find a cofunction with the same value as the given expression.
csc 35°
- In Exercises 31–38, find a cofunction with the same value as the given expression. tan 𝜋 9
Problem 35
Problem 36
Find a cofunction with the same value as the given expression.
tan (𝜋/7)
- In Exercises 31–38, find a cofunction with the same value as the given expression. cos 2𝜋 5
Problem 37
- In Exercises 37–40, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, graph one period of the equation. Then find the following: a. the maximum displacement b. the frequency c. the time required for one cycle d. the phase shift of the motion. d = 3 cos(πt + π/2)
Problem 38
- In Exercises 37–40, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, graph one period of the equation. Then find the following: a. the maximum displacement b. the frequency c. the time required for one cycle d. the phase shift of the motion. d = − 1/2 sin(πt/4 − π/2)
Problem 40
- In Exercises 41–43, find the exact value of each of the remaining trigonometric functions of θ. cos θ = 2/5, sin θ < 0
Problem 41
Problem 42
Find the reference angle for each angle.
5π/4
- In Exercises 45–52, graph two periods of each function. y = 2 tan(x − π/6) + 1
Problem 45
Problem 49
Find the reference angle for each angle.
4.7
- In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. tan x = -1
Problem 55
- In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. csc x = 1
Problem 57
Problem 59
Find the reference angle for each angle.
-25π/6
- In Exercises 61–62, use the figures shown to find the bearing from O to A.
Problem 61
- In Exercises 65–66, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in centimeters. In each exercise, find: a. the maximum displacement b. the frequency c. the time required for one cycle. d = 20 cos π/4 t
Problem 65
- In Exercises 67–68, an object is attached to a coiled spring. In Exercise 67, the object is pulled down (negative direction from the rest position) and then released. In Exercise 68, the object is propelled downward from its rest position. Write an equation for the distance of the object from its rest position after t seconds.
Problem 68
- If θ is an acute angle and cos θ = 1/3, find csc (𝜋/2 - θ).
Problem 72