Question 1

The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated
circular function value of .
-Find csc ϴ.

A)  $-\frac{25}{24}$ 
B) $\frac{24}{7}$ 
C) $-\frac{25}{7}$  
D) $\frac{25}{24}$

Accepted Answer

The answer of The figure shows an angle in standard...

Question 2

Solve the problem.
-A ball on the end of a string is spinning around a circle with a radius of 5 centimeters. If in 13 seconds a central angle of $\frac { 1 } { 5 }$ radian has been covered, what is the angular speed of the ball?
A) $\frac { 5 } { 13 }$ radians per sec
B) $\frac { 13 } { 5 }$ radians per sec
C) 65 radians per sec
D) $\frac { 1 } { 65 }$ radian per sec

Accepted Answer

The angular speed, $\omega$, is given by the formula $\omega = \frac{\theta}{t}$, where $\theta$ is the angle in radians and $t$ is the time in seconds. Here, $\theta = \frac{1}{5}$ radian and $t = 13$ seconds, so $\omega = \frac{1/5}{13} = \frac{1}{65}$ radian per sec.

Question 3

Find the corresponding angle measure in radians.
--150° 
A) $\frac { 2 \pi } { 3 }$
B) $\frac { \pi } { 6 }$
C) $\frac { 7 \pi } { 6 }$
D) $\frac { 5 \pi } { 3 }$

Accepted Answer

The answer of Find the corresponding angle measure in radians.
--150°...

Question 4

Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
-$$\frac { 4 \pi } { 9 }$$
A) $81 ^ { \circ }$
B) $80 ^ { \circ }$
C) $80.5 ^ { \circ }$
D) $79.5 ^ { \circ }$

Accepted Answer

The answer of Convert the radian measure to degrees. Round...

Question 5

Solve the problem.
-Two pulleys of diameters $6 \mathrm {~m}$ and $3 \mathrm {~m}$ are connected by a belt. The larger pulley rotates 42 times per min. Find the angular speed of the smaller pulley.
A) $126 \pi$ radians per min
B) $168 \pi$ radians per min
C) $84 \pi$ radians per min
D) $252 \pi$ radians per min

Accepted Answer

The answer of Solve the problem.
-Two pulleys of diameters \(6...

Question 6

Solve the problem.
-Let angle POQ be designated ϴ. Angles PQR and VRQ are right angles. If ϴ = 43°, find the length of OQ accurate to four decimal places. 
A) $0.9325$
B) $0.7314$
C) $0.6820$
D) $1.3673$

Accepted Answer

The answer of Solve the problem.
-Let angle POQ be designated...

Question 7

Use the formula v = rω to find the value of the missing variable. Give an exact answer unless otherwise indicated.
-v = 218.9 m per sec, Ɯ = 0.63361 radian per sec (Round to four decimal places when necessary.)
A)69.6779 m
B)4.9583 m
C)345.4807 m
D)0.0029 m

Accepted Answer

The answer of Use the formula v = rω to...

Question 8

Find the exact value of s in the given interval that has the given circular function value.
-$$\left[ \pi , \frac { 3 \pi } { 2 } \right] ; \sin \mathrm { s } = - \frac { \sqrt { 3 } } { 2 }$$
A) $s = \frac { 5 \pi } { 4 }$
B) $s = \frac { 7 \pi } { 6 }$
C) $s = \frac { 4 \pi } { 3 }$
D) $\mathrm { s } = \frac { \pi } { 3 }$

Accepted Answer

The sine function equals $-\frac{\sqrt{3}}{2}$ at an angle in the third quadrant, where sine is negative. Among the options, only $s = \frac{4\pi}{3}$ falls in the given interval $\left[\pi, \frac{3\pi}{2}\right]$ and corresponds to the correct sine value.

Question 9

Convert the degree measure to radians, correct to four decimal places. Use $$\text { Use } 3.1416 \text { for } \pi \text {. }$$
-81.4687°
A)1.7219
B)1.5219
C)1.6219
D)1.4219

Accepted Answer

The answer of Convert the degree measure to radians, correct...

Question 10

Use a table or a calculator to evaluate the function. Round to four decimal places.
-csc 0.2475
A)0.9695
B)0.2450
C)1.0314
D)4.0820

Accepted Answer

The cosecant function is the reciprocal of the sine function. To find csc(0.2475), we first find sin(0.2475) and then take the reciprocal. Using a calculator, sin(0.2475) ≈ 0.2450, so csc(0.2475) = 1/sin(0.2475) ≈ 1/0.2450 ≈ 4.0820.

Question 11

Find the exact circular function value.
-$\cot \frac { - 11 \pi } { 6 }$
A) $- \frac { \sqrt { 3 } } { 3 }$
B) $\sqrt { 3 }$
C) $\frac { \sqrt { 3 } } { 3 }$
D) $- \sqrt { 3 }$

Accepted Answer

The cotangent function is periodic with a period of $\pi$, so $\cot \frac{-11\pi}{6} = \cot (\frac{-11\pi}{6} + 2\pi) = \cot \frac{\pi}{6}$. The cotangent of $\frac{\pi}{6}$ is $\sqrt{3}$.

Question 12

Suppose an arc of length $s$ lies on the unit circle $x ^ { 2 } + y ^ { 2 } = 1$, starting at point $( 1,0 )$ and terminating at the point ( $x$, $y$ ). Use a calculator to find the approximate coordinates $( x , y )$. Round coordinates to four decimal places when appropriate.

The unit circle $x ^ { 2 } + y ^ { 2 } = 1$
-s = 5.6
A)(-0.7756, °°°)
B)(-0.6313, 0.7756)
C)(0.7756, -0.6313)
D)(-0.7756, -0.6313)

Accepted Answer

The correct answer is C because s = 5.6 is in the fourth quadrant, and the sine of an angle in the fourth quadrant is positive and the cosine is negative.

Question 13

$$\text { Find the value of s in the interval } [ 0 , \pi / 2 ] \text { that makes the statement true. Round to four decimal places. }$$
-sec s = 3.7404
A)0.2706
B)1.8414
C)1.3002
D)4.9830

Accepted Answer

The secant function is the reciprocal of the cosine function. Therefore, sec(s) = 3.7404 implies cos(s) = 1/3.7404. Calculating this gives s ≈ 1.3002 radians.

Question 14

Use a table or a calculator to evaluate the function. Round to four decimal places.
-cot 0.2258
A)0.9746
B)0.2297
C)1.0260
D)4.3532

Accepted Answer

The cotangent of an angle is the reciprocal of its tangent. Using a calculator to find the tangent of 0.2258 radians and then taking the reciprocal gives approximately 4.3532.

Question 15

Convert the degree measure to radians. Leave answer as a multiple of π.
-1800° 
A) $10 \pi$
B) $\frac { 19 \pi } { 2 }$
C) $\frac { 21 \pi } { 2 }$
D) $20 \pi$

Accepted Answer

To convert degrees to radians, use the conversion factor $\frac{\pi}{180}$. Thus, $1800° \times \frac{\pi}{180} = 10\pi$.

Question 16

The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated
circular function value of .
-Find $\cot \theta$

A) $\frac { 24 } { 7 }$
B) $\frac { 7 } { 24 }$
C) $- \frac { 7 } { 24 }$
D) $- \frac { 24 } { 7 }$

Accepted Answer

The answer of The figure shows an angle in standard...

Question 17

Solve the problem.
-A circular sector has an area of $486 \mathrm { ft } ^ { 2 }$. The radius of the circle is 9 feet. What is the arc length of the sector?
A) $110 \mathrm { ft }$
B) $12 \mathrm { ft }$
C) $108 \mathrm { ft }$
D) $54 \mathrm { ft }$

Accepted Answer

The area of a sector of a circle is given by the formula $A = \frac{1}{2} r^2 \theta$, where $A$ is the area, $r$ is the radius, and $\theta$ is the central angle in radians. Given $A = 486 \, \text{ft}^2$ and $r = 9 \, \text{ft}$, we can solve for $\theta$. First, rearrange the formula to solve for $\theta$: $\theta = \frac{2A}{r^2}$. Substituting the given values, $\theta = \frac{2 \times 486}{9^2} = \frac{972}{81} = 12 \, \text{radians}$. The arc length $L$ of a sector is given by $L = r\theta$. Substituting $r = 9 \, \text{ft}$ and $\theta = 12$, we get $L = 9 \times 12 = 108 \, \text{ft}$.

Question 18

Approximate the length using the formula for arc length. Round to the nearest meter.
-A tree 460 m away subtends an angle of 2°. Find the height of the tree.
A)32 m
B)13 m
C)16 m
D)35 m

Accepted Answer

The formula for arc length is $s = r\theta$, where $s$ is the arc length, $r$ is the radius (or distance to the object), and $\theta$ is the angle in radians. First, convert 2° to radians: $2° \times \frac{\pi}{180°} = \frac{\pi}{90}$. Then, use the formula with $r = 460$ m and $\theta = \frac{\pi}{90}$ radians to find $s$: $s = 460 \times \frac{\pi}{90} \approx 16$ m.

Question 19

Convert the degree measure to radians. Leave answer as a multiple of $\pi .$
--45° 
A) $- \frac { \pi } { 4 }$
B) $- \frac { \pi } { 5 }$
C) $- \frac { \pi } { 3 }$
D) $- \frac { \pi } { 6 }$

Accepted Answer

To convert degrees to radians, multiply by $\frac{\pi}{180}$. Thus, $-45° \times \frac{\pi}{180} = -\frac{\pi}{4}$.

Question 20

Solve the problem.
-Let angle POQ be designated ϴ. Angles PQR and VRQ are right angles. If ϴ = 54°, find the length of VR accurate to four decimal places. 
A) $0.5878$
B) $1.3764$
C) $0.8090$
D) $0.7265$

Accepted Answer

The answer of Solve the problem.
-Let angle POQ be designated...

Quiz 3: Radian Measure and the Unit Circle

Solve the problem. -A ball on the end of a string is spinning around a circle with a radius of 5 centimeters. If in 13 seconds a central angle of $\frac { 1 } { 5 }$ radian has been covered, what is the angular speed of the ball?

Find the corresponding angle measure in radians. --150° $Find the corresponding angle measure in radians. --150° A) \frac { 2 \pi } { 3 } B) \frac { \pi } { 6 } C) \frac { 7 \pi } { 6 } D) \frac { 5 \pi } { 3 }$

Convert the radian measure to degrees. Round to the nearest hundredth if necessary. - $\frac { 4 \pi } { 9 }$

Solve the problem. -Two pulleys of diameters $6 \mathrm {~m}$ and $3 \mathrm {~m}$ are connected by a belt. The larger pulley rotates 42 times per min. Find the angular speed of the smaller pulley.

Solve the problem. -Let angle POQ be designated ϴ. Angles PQR and VRQ are right angles. If ϴ = 43°, find the length of OQ accurate to four decimal places.

Use the formula v = rω to find the value of the missing variable. Give an exact answer unless otherwise indicated. -v = 218.9 m per sec, Ɯ = 0.63361 radian per sec (Round to four decimal places when necessary.)

Find the exact value of s in the given interval that has the given circular function value. - $\left[ \pi , \frac { 3 \pi } { 2 } \right] ; \sin \mathrm { s } = - \frac { \sqrt { 3 } } { 2 }$

Convert the degree measure to radians, correct to four decimal places. Use $\text { Use } 3.1416 \text { for } \pi \text {. }$ -81.4687°

Use a table or a calculator to evaluate the function. Round to four decimal places. -csc 0.2475

Find the exact circular function value. - $\cot \frac { - 11 \pi } { 6 }$

$\text { Find the value of s in the interval } [ 0 , \pi / 2 ] \text { that makes the statement true. Round to four decimal places. }$ -sec s = 3.7404

Use a table or a calculator to evaluate the function. Round to four decimal places. -cot 0.2258

Convert the degree measure to radians. Leave answer as a multiple of π. -1800°

Solve the problem. -A circular sector has an area of $486 \mathrm { ft } ^ { 2 }$ . The radius of the circle is 9 feet. What is the arc length of the sector?

Approximate the length using the formula for arc length. Round to the nearest meter. -A tree 460 m away subtends an angle of 2°. Find the height of the tree.

Convert the degree measure to radians. Leave answer as a multiple of $\pi .$ --45°

Solve the problem. -Let angle POQ be designated ϴ. Angles PQR and VRQ are right angles. If ϴ = 54°, find the length of VR accurate to four decimal places.

Quiz 3: Radian Measure and the Unit Circle

The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated circular function value of . -Find csc ϴ.

Solve the problem. -A ball on the end of a string is spinning around a circle with a radius of 5 centimeters. If in 13 seconds a central angle of 15\frac { 1 } { 5 }51​ radian has been covered, what is the angular speed of the ball?

Find the corresponding angle measure in radians. --150°

Convert the radian measure to degrees. Round to the nearest hundredth if necessary. - 4π9\frac { 4 \pi } { 9 }94π​

Solve the problem. -Two pulleys of diameters 6 m6 \mathrm {~m}6 m and 3 m3 \mathrm {~m}3 m are connected by a belt. The larger pulley rotates 42 times per min. Find the angular speed of the smaller pulley.