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3. Unit Circle

Trigonometric Functions on the Unit Circle

3. Unit Circle

Trigonometric Functions on the Unit Circle

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Sine, Cosine, & Tangent on the Unit Circle

Sine, Cosine, & Tangent on the Unit Circle

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Evaluating Trig Functions of any Angle

Evaluating Trig Functions of any Angle

Carolee Pederson

6 trig functions given any coordinates

6 trig functions given any coordinates

Trig Functions of Any Angle

Trig Functions of Any Angle

Jan-Marie Meadows

Evaluating trig functions of any angle in standard position

Evaluating trig functions of any angle in standard position

Practice this topic

Multiple Choice

Find the sine, cosine, and tangent of each angle using the unit circle.

$\theta=-1.18$ rad, $\left(\frac{5}{13},-\frac{12}{13}\right)$

Multiple Choice

Find the sine, cosine, and tangent of each angle using the unit circle.

$\theta=225\degree,\left(-\frac{\sqrt2}{2},-\frac{\sqrt2}{2}\right)$

Textbook Question

Find exact values of the six trigonometric functions for each angle A.

Textbook Question

CONCEPT PREVIEW Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all. I II 1. A. √3 2. B. 1 3. tan 45° C. ½ 4. D. √3 5. 2 6. E. 2√3 3 F. √3 3 G. 2 H. √2 2 I. √2

Textbook Question

Find each exact function value. See Example 2. sin 7π/6

Textbook Question

Graph each function over a one-period interval.

y = - (1/2) csc (x + π/2)

Textbook Question

CONCEPT PREVIEW Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all. I II 1. A. √3 2. B. 1 3. C. ½ 4. D. √3 5. csc 60° 2 6. E. 2√3 3 F. √3 3 G. 2 H. √2 2 I. √2

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. cos(3θ + 11°) = sin( 7θ + 40°) 5 10

Textbook Question

Find exact values or expressions for sin A, cos A, and tan A. See Example 1.

Textbook Question

Find exact values or expressions for sin A, cos A, and tan A. See Example 1.

Textbook Question

Determine whether each statement is true or false. If false, tell why. tan 60° ≥ cot 40°

Textbook Question

Determine whether each statement is true or false. If false, tell why. csc 22° ≤ csc 68°

Textbook Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. a = 5, b = 12

Textbook Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. a = 6, c = 7

Textbook Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. b = 8, c = 11

Textbook Question

In Exercises 17–20, θ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and cot θ. Where necessary, rationalize denominators. sin θ = 3/5, cos θ = 4/5

Textbook Question

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown side length. Then find exact values of the six trigonometric functions for angle B. Rationalize denominators when applicable. See Example 1. a = √2, c = 2

Textbook Question

In Exercises 17–20, θ is an acute angle and sin θ and cos θ are given. Use identities to find tan θ, csc θ, sec θ, and cot θ. Where necessary, rationalize denominators. __ sin θ = 6, cos θ = √13 7 7

Textbook Question

In Exercises 21–24, θ is an acute angle and sin θ is given. Use the Pythagorean identity sin²θ + cos²θ = 1 to find cos θ. sin θ = 6/7

Textbook Question

Write each function in terms of its cofunction. Assume all angles involved are acute angles. See Example 2. sin 45°

Textbook Question

In Exercises 21–24, θ is an acute angle and sin θ is given. Use the Pythagorean identity sin²θ + cos²θ = 1 to find cos θ. __ sin θ = √39 8

Textbook Question

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sin 37° csc 37°

Textbook Question

Write each function in terms of its cofunction. Assume all angles involved are acute angles. See Example 2. tan 25.4°

Textbook Question

In Exercises 28–29, find a cofunction with the same value as the given expression. sin 70°

Textbook Question

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

Textbook Question

Write each function in terms of its cofunction. Assume all angles involved are acute angles. See Example 2. cos(θ + 20°)

Textbook Question

In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sec² 23° - tan² 23°

Textbook Question

In Exercises 31–38, find a cofunction with the same value as the given expression. sin 7°

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. tan α = cot(α + 10°)

Textbook Question

In Exercises 31–38, find a cofunction with the same value as the given expression. csc 25°

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. sin(2θ + 10°) = cos(3θ - 20°)

Textbook Question

In Exercises 31–38, find a cofunction with the same value as the given expression. tan 𝜋 9

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. cot(5θ + 2°) = tan(2θ + 4°)

Textbook Question

In Exercises 31–38, find a cofunction with the same value as the given expression. cos 2𝜋 5

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. cos(2θ + 50°) = sin(2θ - 20°)

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. sec(3β + 10°) = csc(β + 8°)

Textbook Question

Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. csc(β + 40°) = sec(β - 20°)

Textbook Question

In Exercises 41–43, find the exact value of each of the remaining trigonometric functions of θ. cos θ = 2/5, sin θ < 0

Textbook Question

Determine whether each statement is true or false. See Example 4. tan 28° ≤ tan 40°

Textbook Question

Determine whether each statement is true or false. See Example 4. cos 28° < sin 28° (Hint: sin 28° = cos 62°)

Textbook Question

Determine whether each statement is true or false. See Example 4. cot 30° < tan 40°

Textbook Question

Determine whether each statement is true or false. See Example 4. csc 20° < csc 30°

Textbook Question

Give the exact value of each expression. See Example 5. tan 30°

Textbook Question

Give the exact value of each expression. See Example 5. sin 30°

Textbook Question

Give the exact value of each expression. See Example 5. cos 30°

Textbook Question

Give the exact value of each expression. See Example 5. sec 45°

Textbook Question

Give the exact value of each expression. See Example 5. cot 45°

Textbook Question

Give the exact value of each expression. See Example 5. csc 60°

Textbook Question

Concept Check Work each problem. Find the equation of the line that passes through the origin and makes a 30° angle with the x-axis.

Textbook Question

Concept Check Work each problem. What angle does the line y = √3x make with the positive x-axis?

Textbook Question

Find the exact value of the variables in each figure.

Textbook Question

Find the exact value of the variables in each figure.

Textbook Question

Find a formula for the area of each figure in terms of s.

Textbook Question

Find a formula for the area of each figure in terms of s.

Textbook Question

In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

Textbook Question

In Exercises 1–4, a point P(x, y) is shown on the unit circle corresponding to a real number t. Find the values of the trigonometric functions at t.

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined. sin 𝜋/3

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. csc 4𝜋/3

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. sec 5𝜋/3

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. cos 3𝜋/2

Textbook Question

In Exercises 5–18, the unit circle has been divided into twelve equal arcs, corresponding to t-values of 0, 𝜋, 𝜋, 𝜋, 2𝜋, 5𝜋, 𝜋, 7𝜋, 4𝜋, 3𝜋, 5𝜋, 11𝜋, and 2𝜋. 6 3 2 3 6 6 3 2 3 6 Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.

In Exercises 11–18, continue to refer to the figure at the bottom of the previous page. tan 3𝜋/2

Textbook Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function. b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. cos (-𝜋/6)

Textbook Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function. b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. cos 𝜋/3

Textbook Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function. b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. sin 5𝜋/6

Textbook Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function. b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. tan 5𝜋/3

Textbook Question

In Exercises 19–24, a. Use the unit circle shown for Exercises 5–18 to find the value of the trigonometric function. b. Use even and odd properties of trigonometric functions and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. tan (-11𝜋/6)

Textbook Question

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.

Textbook Question

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.

Textbook Question

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.

Textbook Question

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.

Textbook Question

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

Textbook Question

Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

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Textbook Question

Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

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Textbook Question

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.

s = 2π

Textbook Question

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.

s = ―π

Textbook Question

Find each exact function value. See Example 2. cos (―4π/3)

Textbook Question

Find each exact function value. See Example 2.

cos 7π/4

Textbook Question

Find each exact function value. See Example 2.

sin (4π/3)

Textbook Question

Find each exact function value. See Example 2.

sec 23π/6

Textbook Question

Find each exact function value. See Example 2.

tan 5π/6

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. See Example 3. sin 0.6109

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. See Example 3. cos (-1.1519)

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. See Example 3. tan 4.0203

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. See Example 3. csc (―9.4946)

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. See Example 3.

sec 2.8440

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. See Example 3.

cot 6.0301

Textbook Question

Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)

cos 2

Textbook Question

Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)

sin ( ―1)

Textbook Question

Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)

sin 5

Textbook Question

Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)

cos 6

Textbook Question

Without using a calculator, decide whether each function value is positive or negative. (Hint: Consider the radian measures of the quadrantal angles, and remember that π ≈ 3.14.)

tan 6.29

Textbook Question

Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.

tan s = 0.2126

Textbook Question

Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.

cos s = 0.7826

Textbook Question

Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.

sin s = 0.9918

Textbook Question

Find the approximate value of s, to four decimal places, in the interval [0 , π/2] that makes each statement true.

sec s = 1.0806

Textbook Question

Find the exact value of s in the given interval that has the given circular function value.

[π/2, π] ; sin s = 1/2

Textbook Question

Find the exact value of s in the given interval that has the given circular function value.

[π, 3π/2] ; tan s = √3

Textbook Question

Find the exact value of s in the given interval that has the given circular function value.

[3π/2, 2π] ; tan s = -1

Textbook Question

Find the exact values of s in the given interval that satisfy the given condition.

[0, 2π) ; sin s = -√3 / 2

Textbook Question

Find the exact values of s in the given interval that satisfy the given condition.

[0 , 2π) ; cos² s = 1/2

Textbook Question

Find the exact values of s in the given interval that satisfy the given condition.

[-2π , π) ; 3 tan² s = 1

Textbook Question

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.

s = 2.5

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Textbook Question

Suppose an arc of length s lies on the unit circle x² + y² = 1, starting at the point (1, 0) and terminating at the point (x, y). (See Figure 12, repeated below.) Use a calculator to find the approximate coordinates for (x, y) to four decimal places.

s = ―7.4

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Textbook Question

For each value of s, use a calculator to find sin s and cos s, and then use the results to decide in which quadrant an angle of s radians lies.

s = 51

Textbook Question

For each value of s, use a calculator to find sin s and cos s, and then use the results to decide in which quadrant an angle of s radians lies.

s = 65

Textbook Question

Find each exact function value.

tan π/3

Textbook Question

Find each exact function value.

sin ( ―5π/6)

Textbook Question

Find each exact function value.

csc ( ―11π/6)

Textbook Question

Without using a calculator, determine which of the two values is greater.

tan 1 or tan 2

Textbook Question

Without using a calculator, determine which of the two values is greater.

cos 2 or sin 2

Textbook Question

Find a calculator approximation to four decimal places for each circular function value.

sin 1.0472

Textbook Question

Find a calculator approximation to four decimal places for each circular function value. cos (-0.2443)

Textbook Question

Find a calculator approximation to four decimal places for each circular function value.

sec 7.3159

Textbook Question

Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.

cos s = 0.9250

Textbook Question

Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.

sin s = 0.4924

Textbook Question

Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.

cot s = 0.5022

Textbook Question

Find the exact value of s in the given interval that has the given circular function value.

[ 0, π/2] ; cos s = √2/2

Textbook Question

Find the exact value of s in the given interval that has the given circular function value.

[ π , 3π/2] ; sec s = ―2√3/3

Textbook Question

Find the exact values of (a) sin s, (b) cos s, and (c) tan s for each real number s. See Example 1.

s = π/2

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