3. Unit Circle

Identify the quadrant that the given angle is located in.

$\frac{7\pi}{4}$ radians

Identify the quadrant that the given angle is located in.

$\frac{\pi}{7}$ radians

Identify the quadrant that the given angle is located in.

$\frac{2\pi}{3}$ radians

Identify the quadrant that the given angle is located in.

$\frac{6\pi}{5}$ radians

Test whether the point is on the unit circle by plugging it into the equation, $x^2+y^2=1$.

$\left(\frac{-\sqrt2}{2},\frac{-\sqrt2}{2}\right)$

Use the formula ω = θ/t to find the value of the missing variable.

ω = 2π/3 radians per sec, t = 3 sec

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

Use the formula ω = θ/t to find the value of the missing variable.

ω = 0.91 radian per min, t = 8.1 min

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

Use the formula ω = θ/t to find the value of the missing variable.

θ = 3π/4 radians, t = 8 sec

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

Use the formula ω = θ/t to find the value of the missing variable.

θ = 3.871 radians, t = 21.47 sec

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

Use the formula ω = θ/t to find the value of the missing variable.

θ = 2π/9 radian , ω = 5π/27 radian per min

Find each exact function value. See Example 2.

tan 3π/4

Use the formula v = r ω to find the value of the missing variable.

r = 12 m , ω = 2π/3 radians per sec

Find each exact function value. See Example 2.

csc 11π/6

Use the formula v = r ω to find the value of the missing variable.

v = 9 m per sec , r = 5 m

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

Use the formula v = r ω to find the value of the missing variable.

v = 12 m per sec, ω = 3π/2 radians per sec

Find each exact function value. See Example 2.

cos 7π/4

The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.

r = 6 cm, ω = π/3 radians per sec, t = 9 sec

Find each exact function value. See Example 2.

sin (4π/3)

The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.

s = 6π cm, r = 2 cm, ω = π/4 radian per sec

Find each exact function value. See Example 2.

sec 23π/6

The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.

s = 3π/4 km, r = 2 km, t = 4 sec

Find each exact function value. See Example 2.

tan 5π/6

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

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

Find each exact function value.

tan π/3

Find the angular speed ω for each of the following.

a gear revolving 300 times per min

Find the angular speed ω for each of the following.

a wind turbine with blades turning at a rate of 15 revolutions per minute

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

Find each exact function value.

sin ( ―5π/6)

Find the linear speed v for each of the following.

the tip ​​of the minute hand of a clock, if the hand is 7 cm long

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

Find each exact function value.

csc ( ―11π/6)

Find the linear speed v for each of the following.

a point on the edge of a flywheel of radius 2 m, rotating 42 times per min

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

sec 2.8440

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

tan 1 or tan 2

Find the linear speed v for each of the following.

the tip of a propeller 3 m long, rotating 500 times per min (Hint: r = 1.5 m)

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

cot 6.0301

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

cos 2 or sin 2

Find the linear speed v for each of the following.

a point on the equator moving due to Earth's rotation, if the radius is 3960 mi

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

sin 1.0472

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

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

sec 7.3159

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

A thread is being pulled off a spool at the rate of 59.4 cm per sec. Find the radius of the spool if it makes 152 revolutions per min.

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

The propeller of a 90-horsepower outboard motor at full throttle rotates at exactly 5000 revolutions per min. Find the angular speed of the propeller in radians per second.

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

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

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)

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

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

[ 0, π/2] ; cos s = √2/2

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

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

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

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

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

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

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

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

[π/2, π] ; sin s = 1/2

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

[π, 3π/2] ; tan s = √3

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

[3π/2, 2π] ; tan s = -1

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

[0, 2π) ; sin s = -√3/ 2

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

[0, 2π) ; sin s = -√3 / 2

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

[0 , 2π) ; cos² s = 1/2

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

[-2π , π) ; 3 tan² s = 1

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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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

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

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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