Question 1

Use a table or a calculator to evaluate the function. Round to four decimal places.
-sec 0.2020
A)0.9797
B)1.0208
C)0.2048
D)0.2006

Accepted Answer

The secant of an angle is the reciprocal of the cosine of that angle. Using a calculator to find the cosine of 0.2020 radians and then taking the reciprocal gives approximately 1.0208.

Question 2

Find the area of a sector of a circle having radius r and central angle . If necessary, express the answer to the nearest
tenth.
-$\mathrm { r } = 44.6 \mathrm {~cm} , \theta = \frac { \pi } { 13 }$ radians
A) $5.4 \mathrm {~cm} ^ { 2 }$
B) $240.4 \mathrm {~cm} ^ { 2 }$
C) $76.5 \mathrm {~cm} ^ { 2 }$
D) $480.8 \mathrm {~cm} ^ { 2 }$

Accepted Answer

The area of a sector of a circle is given by the formula $A = \frac{1}{2} r^2 \theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Substituting the given values, $A = \frac{1}{2} \times (44.6)^2 \times \frac{\pi}{13} \approx 240.4 \mathrm{~cm}^2$.

Question 3

Find the specified quantity.
-$$\text { Is it correct to say that the value of } \tan 30 = \frac { \sqrt { 3 } } { 3 } \text { ? Explain your answer. }$$

Accepted Answer

No, it is not correct because tan 30 means the tangent of 30 radians, not the tangent of 30 degrees. It would be correct to say that the value of $\tan 30 ^ { \circ } = \frac { \sqrt { 3 } } { 3 }$.

Question 4

Solve the problem.
-Find $\omega$ for a spoke on a bike tire revolving 60 times per minute.
A) $60 \pi$ radians per min
B) $\frac { \pi } { 120 }$ radians per min
C) $\frac { \pi } { 60 }$ radians per min
D) $120 \pi$ radians per min

Accepted Answer

The angular velocity $\omega$ is calculated by multiplying the number of revolutions per minute by $2\pi$ radians (since one revolution is $2\pi$ radians). Thus, $\omega = 60 \times 2\pi = 120\pi$ radians per minute.

Question 5

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

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

Accepted Answer

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

Question 6

$$\text { Convert the degree measure to radians, correct to four decimal places. Use } 3.1416 \text { for } \pi \text {. }$$
-83.88°
A)1.5640
B)1.4640
C)1.3640
D)1.2640

Accepted Answer

To convert degrees to radians, use the formula radians = degrees × (π / 180). For 83.88°, the calculation is 83.88 × (3.1416 / 180) = 1.4640.

Question 7

Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π.
--160°41´
A)-2.8045
B)-2.7945
C)-2.7745
D)-2.7845

Accepted Answer

-160°41´ = -160° - 41/60° = -160° - 0.6833° = -160.6833°= -160.6833° * π/180° = -2.7945 radians (correct to four decimal places)

Question 8

$$\text { Convert the radian measure to degrees. Give answer using decimal degrees to the nearest hundredth. Use } 3.1416 \text { for }$$
-0.8996
A)52.04°
B)52.54°
C)51.54°
D)50.84°

Accepted Answer

The answer of \[\text { Convert the radian measure to...

Question 9

4
Provide an appropriate response.
-Describe how an angle measure can be converted from radians to degrees.
A) Multiply the degree measure by $\frac { 360 ^ { \circ } } { \pi }$.
B) Multiply the degree measure by $\frac { 180 ^ { \circ } } { \pi }$.
C) Multiply the degree measure by $\frac { \pi } { 90 ^ { \circ } }$.
D) Multiply the degree measure by $\frac { \pi } { 180 ^ { \circ } }$.

Accepted Answer

To convert an angle from radians to degrees, multiply the radian measure by $\frac { 180 ^ { \circ } } { \pi }$. This is because $\pi$ radians is equal to 180 degrees.

Question 10

Solve the problem.
-Find $\omega$ for the minute hand of a clock.
A) $\frac { \pi } { 60 }$ radians per min
B) $\frac { \pi } { 6 }$ radians per min
C) $\frac { \pi } { 30 }$ radians per min
D) $30 \pi$ radians per min

Accepted Answer

The minute hand of a clock completes one full rotation (360 degrees or $2\pi$ radians) in 60 minutes. Therefore, its angular velocity, $\omega$, is $\frac{2\pi}{60} = \frac{\pi}{30}$ radians per minute.

Question 11

Find the specified quantity.
-If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central
angle changed?

Accepted Answer

The answer of Find the specified quantity.
-If the radius of...

Question 12

Find the specified quantity.
-$$\text { Is it correct to say that the value of } \cos 45 = \frac { \sqrt { 2 } } { 2 } \text { ? Explain your answer. }$$

Accepted Answer

The answer of Find the specified quantity.
-\[\text { Is it...

Question 13

Solve the problem.
-Suppose the tip of the minute hand of a clock is 5 in. from the center of the clock. Determine the distance traveled by the tip of the minute hand in $3 \frac { 1 } { 2 }$ hours. Give an exact answer.
A) $30 \pi \mathrm { in }$.
B) $35 \pi \mathrm { in }$.
C) $15 \pi \mathrm { in }$.
D) $20 \pi$ in.

Accepted Answer

The distance traveled by the tip of the minute hand in one complete revolution is the circumference of the circle it traces, which is given by $C = 2\pi r$, where $r$ is the radius of the circle. Here, $r = 5$ inches, so $C = 2\pi(5) = 10\pi$ inches per hour. In $3 \frac{1}{2}$ hours, or 3.5 hours, the distance traveled is $3.5 \times 10\pi = 35\pi$ inches.

Question 14

Solve the problem.
-Suppose the tip of the minute hand of a clock is 15 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 10 minutes. Give an exact answer. 
A) $10 \pi \mathrm { in }$.
B) $5 \pi \mathrm { in }$.
C) $2.5 \pi \mathrm { in }$.
D) $25 \pi \mathrm { in }$.

Accepted Answer

The minute hand completes a full circle (360 degrees) in 60 minutes. In 10 minutes, it covers 1/6 of the circle. The circumference of the circle is $2 \pi \times 15 = 30 \pi$ inches. Therefore, the distance traveled in 10 minutes is $(1/6) \times 30 \pi = 5 \pi$ inches.

Question 15

Find the exact value of s in the given interval that has the given circular function value.
-$$\left[ \pi , \frac { 3 \pi } { 2 } \right] ; \tan \mathrm { s } = 1$$
A) $\mathrm { s } = \frac { 5 \pi } { 4 }$
B) $s = \frac { 4 \pi } { 3 }$
C) $s = \frac { 7 \pi } { 6 }$
D) $s = \frac { \pi } { 4 }$

Accepted Answer

The tangent function equals 1 at an angle of $\frac{\pi}{4}$ radians. However, in the given interval $\left[ \pi , \frac { 3 \pi } { 2 } \right]$, the angle that also has a tangent of 1 is $\frac { 5 \pi } { 4 }$, because it is in the third quadrant where both sine and cosine are negative, making their ratio positive.

Question 16

Find the exact value without using a calculator.
-$\cot \left( \frac { - 5 \pi } { 6 } \right)$
A) $- \sqrt { 3 }$
B) $- \frac { \sqrt { 3 } } { 3 }$
C) $\sqrt { 3 }$
D) $\frac { \sqrt { 3 } } { 3 }$

Accepted Answer

The answer of Find the exact value without using a...

Question 17

Find the specified quantity.
-Find the value of the arc length when ϴ (the central Angle)is given in degrees instead of
radians.

Accepted Answer

The answer of Find the specified quantity.
-Find the value of...

Question 18

Find the exact circular function value.
-$\cot \pi$
A) $- 1$
B) undefined
C) 0
D) 1

Accepted Answer

The cotangent function, $\cot\theta$, is defined as $\frac{1}{\tan\theta}$ or $\frac{\cos\theta}{\sin\theta}$. For $\theta = \pi$, $\sin\pi = 0$, making the denominator of the cotangent function zero. Therefore, $\cot \pi$ is undefined.

Question 19

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

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

Accepted Answer

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

Question 20

For the given value of s, decide in which quadrant an angle of s radians lies by evaluating sin s and cos s.
-s = 47
A)III
B)I
C)IV
D)II

Accepted Answer

The answer of For the given value of s, decide...

Quiz 3: Radian Measure and the Unit Circle

Use a table or a calculator to evaluate the function. Round to four decimal places. -sec 0.2020

Find the area of a sector of a circle having radius r and central angle . If necessary, express the answer to the nearest tenth. - $\mathrm { r } = 44.6 \mathrm {~cm} , \theta = \frac { \pi } { 13 }$ radians

Find the specified quantity. - $\text { Is it correct to say that the value of } \tan 30 = \frac { \sqrt { 3 } } { 3 } \text { ? Explain your answer. }$

Solve the problem. -Find $\omega$ for a spoke on a bike tire revolving 60 times per minute.

$\text { Convert the degree measure to radians, correct to four decimal places. Use } 3.1416 \text { for } \pi \text {. }$ -83.88°

Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π. --160°41´

$\text { Convert the radian measure to degrees. Give answer using decimal degrees to the nearest hundredth. Use } 3.1416 \text { for }$ -0.8996

4 Provide an appropriate response. -Describe how an angle measure can be converted from radians to degrees.

Solve the problem. -Find $\omega$ for the minute hand of a clock.

Find the specified quantity. -If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

Find the specified quantity. - $\text { Is it correct to say that the value of } \cos 45 = \frac { \sqrt { 2 } } { 2 } \text { ? Explain your answer. }$

Solve the problem. -Suppose the tip of the minute hand of a clock is 5 in. from the center of the clock. Determine the distance traveled by the tip of the minute hand in $3 \frac { 1 } { 2 }$ hours. Give an exact answer.

Solve the problem. -Suppose the tip of the minute hand of a clock is 15 inches from the center of the clock. Determine the distance traveled by the tip of the minute hand in 10 minutes. Give an exact answer.

Find the exact value of s in the given interval that has the given circular function value. - $\left[ \pi , \frac { 3 \pi } { 2 } \right] ; \tan \mathrm { s } = 1$

Find the exact value without using a calculator. - $\cot \left( \frac { - 5 \pi } { 6 } \right)$

Find the specified quantity. -Find the value of the arc length when ϴ (the central Angle)is given in degrees instead of radians.

Find the exact circular function value. - $\cot \pi$

For the given value of s, decide in which quadrant an angle of s radians lies by evaluating sin s and cos s. -s = 47

Quiz 3: Radian Measure and the Unit Circle

Use a table or a calculator to evaluate the function. Round to four decimal places. -sec 0.2020

Find the area of a sector of a circle having radius r and central angle . If necessary, express the answer to the nearest tenth. - r=44.6 cm,θ=π13\mathrm { r } = 44.6 \mathrm {~cm} , \theta = \frac { \pi } { 13 }r=44.6 cm,θ=13π​ radians

Solve the problem. -Find ω\omegaω for a spoke on a bike tire revolving 60 times per minute.

The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated circular function value of . - Find cos⁡θ. \text { Find } \cos \theta \text {. } Find cosθ.

Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π. --160°41´

4 Provide an appropriate response. -Describe how an angle measure can be converted from radians to degrees.

Solve the problem. -Find ω\omegaω for the minute hand of a clock.

Find the specified quantity. -If the radius of a circle is doubled, how is the length of the arc intercepted by a fixed central angle changed?

Solve the problem. -Suppose the tip of the minute hand of a clock is 5 in. from the center of the clock. Determine the distance traveled by the tip of the minute hand in 3123 \frac { 1 } { 2 }321​ hours. Give an exact answer.