Radians and Degrees Practice Questions — SAT

Question 1Question →

A robotic arm rotates counterclockwise by $210^\circ$ and then rotates clockwise by $75^\circ$ . What is the net counterclockwise rotation of the robotic arm, in radians?

\frac{3\pi}{4}

\frac{4\pi}{3}

\frac{29\pi}{36}

\frac{3\pi}{5}

Show answer & explanation

Answer: $\frac{3\pi}{4}$

Answer

The correct answer is

\frac{3\pi}{4}

radians.

The correct answer is determined by first finding the net angle of rotation. Subtracting the

75^\circ

clockwise rotation from the

210^\circ

counterclockwise rotation yields a net rotation of

135^\circ

counterclockwise. Converting this angle to radians is done by multiplying

135^\circ

by

\frac{\pi}{180^\circ}

, resulting in

\frac{135\pi}{180}

radians. Dividing both the numerator and denominator by 45 simplifies the fraction to

\frac{3\pi}{4}

radians.

Step-by-Step Solution

1

Calculate the net counterclockwise rotation in degrees.

210^\circ - 75^\circ = 135^\circ

Since counterclockwise rotation is positive and clockwise rotation is negative, we subtract the clockwise rotation from the counterclockwise rotation.

2

Convert the angle from degrees to radians.

135 \times \frac{\pi}{180} = \frac{135\pi}{180}

radians

To convert degrees to radians, multiply the degree measure by the conversion factor

\frac{\pi}{180^\circ}

.

3

Simplify the fraction to its lowest terms.

\frac{3\pi}{4}

radians

Divide both the numerator and the denominator by their greatest common divisor, which is 45.

Key Concept

To convert an angle from degrees to radians, multiply the degree measure by

\frac{\pi}{180^\circ}

and simplify the resulting fraction.

Alternative Method

Alternatively, convert each individual rotation to radians first, then subtract. Convert

210^\circ

to radians:

210 \times \frac{\pi}{180} = \frac{7\pi}{6}

radians. Convert

75^\circ

to radians:

75 \times \frac{\pi}{180} = \frac{5\pi}{12}

radians. Subtract the clockwise rotation from the counterclockwise rotation:

\frac{7\pi}{6} - \frac{5\pi}{12} = \frac{14\pi}{12} - \frac{5\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4}

radians.

Estimated Time:1m 30s

Question 2Question →

An angle has a measure of $72^\circ$ . What is the measure of this angle in radians?

\frac{2\pi}{5}

\frac{3\pi}{5}

\frac{5\pi}{2}

\frac{\pi}{5}

Show answer & explanation

Answer: $\frac{2\pi}{5}$

Answer

The correct answer is

\frac{2\pi}{5}

radians.

To convert an angle from degrees to radians, multiply the degree measure by

\frac{\pi}{180}

. Doing so for

72^\circ

yields

\frac{72\pi}{180}

. Dividing the numerator and denominator by their greatest common divisor, 36, simplifies the fraction to

\frac{2\pi}{5}

.

Step-by-Step Solution

1

Set up the conversion from degrees to radians by multiplying the given degree measure by

\frac{\pi}{180}

.

\theta = 72 \times \frac{\pi}{180} = \frac{72\pi}{180}

The conversion factor from degrees to radians is

\frac{\pi \text{ radians}}{180^\circ}

.

2

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, 36.

\frac{72 \div 36}{180 \div 36}\pi = \frac{2\pi}{5}

Simplifying the fraction expresses the angle in standard reduced radian form.

Key Concept

Converting degrees to radians

Question 3Question →

A wheel rotates through a central angle of $\frac{11\pi}{18}$ radians. What is the measure of this angle in degrees?

Show answer & explanation

Answer: 110

Answer

110

To convert radians to degrees, multiply the angle by

\frac{180}{\pi}

. Thus,

\frac{11\pi}{18} \times \frac{180}{\pi} = 110

. The measure of the angle is 110 degrees.

Step-by-Step Solution

1

Identify the conversion relationship between radians and degrees.

Multiply the angle in radians by

\frac{180}{\pi}

to convert to degrees.

Since

\pi

radians is equal to

180

degrees, the conversion factor is

\frac{180}{\pi}

.

2

Multiply the given radian measure of

\frac{11\pi}{18}

by the conversion factor.

110

Applying the conversion factor simplifies the expression by canceling

\pi

and dividing

180

by

18

to get

10

, which is then multiplied by

11

to get the final degree measure.

Key Concept

To convert an angle from radians to degrees, multiply the angle in radians by

\frac{180}{\pi}

.

Estimated Time:45s

Question 4Question →

If the measure of an angle is $\frac{5\pi}{4}$ radians, what is its measure in degrees?

144^\circ

200^\circ

225^\circ

450^\circ

Show answer & explanation

Answer: $225^\circ$

Answer

225^\circ

To convert an angle from radians to degrees, multiply the radian measure by

\frac{180^\circ}{\pi}

. Multiplying

\frac{5\pi}{4}

by

\frac{180^\circ}{\pi}

simplifies to

\frac{5}{4} \times 180^\circ = 5 \times 45^\circ = 225^\circ

. Therefore, the correct degree measure of the angle is

225^\circ

.

Step-by-Step Solution

1

Set up the conversion from radians to degrees by multiplying the given radian measure by

\frac{180^\circ}{\pi}

.

\frac{5\pi}{4} \times \frac{180^\circ}{\pi}

Since

\pi

radians is equal to

180^\circ

, multiplying by the conversion factor

\frac{180^\circ}{\pi}

converts the units from radians to degrees.

2

Simplify the expression by canceling the

\pi

terms and performing the arithmetic.

225^\circ

The

\pi

in the numerator and denominator cancel out, leaving

\frac{5}{4} \times 180^\circ

. Dividing

180

by

4

gives

45

, and multiplying

5

by

45

yields

225

.

Key Concept

To convert an angle from radians to degrees, multiply the angle in radians by

\frac{180^\circ}{\pi}

.

Alternative Method

Another way to convert is to substitute

180^\circ

directly for

\pi

in the expression:

\frac{5\pi}{4} = \frac{5(180^\circ)}{4} = 5 \times 45^\circ = 225^\circ

.

Estimated Time:45s

Question 5Question →

In the $xy$ -plane, an angle $\theta$ in standard position has its terminal ray intersecting the unit circle at the point $\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$ . The angle is rotated counterclockwise by $15^\circ$ , and then its measure is doubled. The final resulting angle is coterminal with an angle of $\frac{a\pi}{b}$ radians, where $0 \le \frac{a\pi}{b} < 2\pi$ , and $a$ and $b$ are positive integers with no common factors. What is the value of $a + b$ ?

Show answer & explanation

Answer: 5

Answer

5

The correct answer is 5. Starting with the point

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

on the unit circle, the angle in standard position is

225^\circ

(or

\frac{5\pi}{4}

radians). Adding

15^\circ

for the counterclockwise rotation yields

240^\circ

(or

\frac{4\pi}{3}

radians). Doubling this measure gives

480^\circ

(or

\frac{8\pi}{3}

radians). Finding the coterminal angle within

[0, 2\pi)

yields

120^\circ

(or

\frac{2\pi}{3}

radians). In this form,

a=2

and

b=3

, which are positive integers with no common factors, so

a+b = 5

.

Step-by-Step Solution

1

Determine the initial angle of the terminal ray intersecting the unit circle at

\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)

.

The initial angle

\theta

is

225^\circ

(or

\frac{5\pi}{4}

radians).

Since both the

x

- and

y

-coordinates are negative and equal, the angle lies in the third quadrant and forms a

45^\circ

reference angle with the negative

x

-axis, which corresponds to

225^\circ

.

2

Apply a counterclockwise rotation of

15^\circ

to the initial angle.

The new angle is

240^\circ

(or

\frac{4\pi}{3}

radians).

Counterclockwise rotations correspond to adding positive angle measures:

225^\circ + 15^\circ = 240^\circ

.

3

Double the measure of the rotated angle.

The doubled angle is

480^\circ

(or

\frac{8\pi}{3}

radians).

The prompt specifies that the angle's measure is doubled after the rotation:

240^\circ \times 2 = 480^\circ

.

4

Find the coterminal angle within the standard interval

[0, 2\pi)

radians (or

[0^\circ, 360^\circ)

).

The coterminal angle is

120^\circ

(or

\frac{2\pi}{3}

radians).

To bring the angle back into the interval

[0^\circ, 360^\circ)

, subtract

360^\circ

(one full rotation):

480^\circ - 360^\circ = 120^\circ

. In radians,

120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}

radians.

5

Identify

a

and

b

and calculate their sum.

a = 2

,

b = 3

, and

a + b = 5

.

The coterminal angle is expressed as

\frac{a\pi}{b}

in simplest form, so

a=2

and

b=3

, which are positive integers with no common factors.

Key Concept

Converting between radians and degrees, finding coterminal angles, and performing angle transformations.

Estimated Time:3m 0s

Question 6Question →

In the $xy$ -plane, an angle $\theta$ is in standard position. The terminal ray of $\theta$ is rotated counterclockwise by $\frac{7\pi}{6}$ radians, and then rotated clockwise by $135^\circ$ . If the terminal ray of the resulting angle lies on the line $y = -x$ in the fourth quadrant, and the original angle $\theta$ has a measure of $d$ degrees, where $0 \leq d < 360$ , what is the value of $d$ ?

Show answer & explanation

Answer: 240

Answer

The value of

d

is 240.

A terminal ray in the fourth quadrant lying on the line

y = -x

forms a

315^\circ

angle in standard position. The first rotation is counterclockwise by

\frac{7\pi}{6}

radians, which is equivalent to

\frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ

. The second rotation is clockwise by

135^\circ

, representing a decrease of

135^\circ

. Therefore, the total transformation is

\theta + 210^\circ - 135^\circ = 315^\circ

. Solving for

\theta

gives

\theta + 75^\circ = 315^\circ

, which simplifies to

\theta = 240^\circ

. Since

240^\circ

lies in the interval

[0, 360)

, the value of

d

is 240.

Step-by-Step Solution

1

Determine the angle in standard position for a terminal ray on the line

y = -x

in the fourth quadrant.

The terminal ray corresponds to an angle of

315^\circ

(or any angle coterminal with it).

The line

y = -x

in the fourth quadrant makes an angle of

45^\circ

below the positive

x

-axis, which corresponds to

360^\circ - 45^\circ = 315^\circ

in standard position.

2

Convert the counterclockwise rotation from radians to degrees.

\frac{7\pi}{6} \text{ radians} = 210^\circ

.

To convert radians to degrees, multiply the angle in radians by

\frac{180^\circ}{\pi}

.

3

Express the rotations mathematically and set up the equation for

\theta

.

\theta + 210^\circ - 135^\circ = 315^\circ + 360^\circ n

(where

n

is an integer).

In standard position, counterclockwise rotations represent positive changes in angle measure, whereas clockwise rotations represent negative changes in angle measure.

4

Solve for

\theta

and apply the domain restriction

0 \leq d < 360

.

\theta = 240^\circ

, so

d = 240

.

Simplifying the equation gives

\theta + 75^\circ = 315^\circ

, which yields

\theta = 240^\circ

when

n=0

.

Key Concept

Converting angles from radians to degrees, understanding the direction of rotation, and determining standard position angles on the coordinate plane.

Question 7Question →

In the $xy$ -plane, a particle moves along a circular path centered at the origin. The particle starts at the point $(3, 0)$ and travels counterclockwise along the circle. If the particle stops after traveling an arc length of $5\pi$ units, what is the measure of the central angle subtended by the particle's path, in degrees?

150^\circ

225^\circ

300^\circ

600^\circ

Show answer & explanation

Answer: $300^\circ$

Answer

300^\circ

The radius of the circular path is

3

units since it starts at

(3,0)

and is centered at the origin. Using the arc length formula

s = r\theta

where

s = 5\pi

is the arc length and

r = 3

is the radius, the central angle is

\theta = \frac{5\pi}{3}

radians. To convert this angle to degrees, we multiply by

\frac{180^\circ}{\pi}

, which yields

\frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 300^\circ

.

Step-by-Step Solution

1

Determine the radius of the circular path.

r = 3

Since the path is centered at the origin

(0, 0)

and starts at

(3, 0)

, the distance from the center to the path (the radius) is

3

units.

2

Calculate the central angle in radians using the arc length formula.

\theta = \frac{5\pi}{3}

radians

The relationship between arc length

s

, radius

r

, and central angle

\theta

in radians is given by

s = r\theta

. Substituting

s = 5\pi

and

r = 3

gives

5\pi = 3\theta

, which simplifies to

\theta = \frac{5\pi}{3}

.

3

Convert the central angle from radians to degrees.

300^\circ

To convert radians to degrees, multiply the angle in radians by

\frac{180^\circ}{\pi}

. This gives

\theta = \frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 5 \times 60^\circ = 300^\circ

.

Key Concept

Using the arc length formula

s = r\theta

to find an angle in radians and converting that angle to degrees.

Question 8Question →

In the $xy$ -plane, a circle with its center at the origin contains the point $P(6, -2\sqrt{3})$ . Point $P$ is rotated counterclockwise about the origin by an angle of $\frac{5\pi}{6}$ radians to a new position $Q$ . What are the coordinates of point $Q$ ?

(0, -4\sqrt{3})

(-4\sqrt{3}, 0)

(-2\sqrt{3}, 6)

(-6, 2\sqrt{3})

Show answer & explanation

Answer: $(-2\sqrt{3}, 6)$

Answer

The coordinates of point

Q

are

(-2\sqrt{3}, 6)

.

The correct coordinates are found by calculating the circle's radius

R = 4\sqrt{3}

and identifying that the initial point

P(6, -2\sqrt{3})

has an angle of

-\frac{\pi}{6}

radians. Rotating counterclockwise by

\frac{5\pi}{6}

radians results in a new angle of

\frac{2\pi}{3}

radians (

120^\circ

). Using circular trigonometry,

x = R\cos(120^\circ) = -2\sqrt{3}

and

y = R\sin(120^\circ) = 6

, giving the point

(-2\sqrt{3}, 6)

.

Step-by-Step Solution

1

Find the radius

R

of the circle using the coordinates of point

P(6, -2\sqrt{3})

.

R = \sqrt{6^2 + (-2\sqrt{3})^2} = \sqrt{36 + 12} = \sqrt{48} = 4\sqrt{3}

.

The radius of the circle is needed to determine the coordinates of the rotated point

Q

on the same circle.

2

Determine the initial angle

\theta_P

of point

P

in standard position.

Since

\cos(\theta_P) = \frac{6}{4\sqrt{3}} = \frac{\sqrt{3}}{2}

and

\sin(\theta_P) = \frac{-2\sqrt{3}}{4\sqrt{3}} = -\frac{1}{2}

, the point lies in the fourth quadrant with

\theta_P = -\frac{\pi}{6}

radians (or

330^\circ

).

Finding the initial position in standard angle form allows us to apply the rotation.

3

Calculate the new angle

\theta_Q

after the counterclockwise rotation of

\frac{5\pi}{6}

radians.

\theta_Q = -\frac{\pi}{6} + \frac{5\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3}

radians.

Adding the counterclockwise rotation angle to the initial angle gives the new standard position angle of the terminal ray.

4

Convert the new angle to degrees and compute the coordinates of point

Q

.

Converting the angle gives

\frac{2\pi}{3} \times \frac{180^\circ}{\pi} = 120^\circ

. The coordinates of

Q

are

x_Q = 4\sqrt{3} \cos(120^\circ) = -2\sqrt{3}

and

y_Q = 4\sqrt{3} \sin(120^\circ) = 6

.

Converting to degrees allows the use of standard trigonometric values to find the exact coordinates.

Key Concept

Radian-to-degree conversion and application of rotations on a coordinate plane using circular trigonometry.

Alternative Method

Instead of working entirely in radians, convert the coordinates of

P

to degrees first. Since

\tan(\theta_P) = -\frac{\sqrt{3}}{3}

in Quadrant IV,

\theta_P = -30^\circ

. The rotation of

\frac{5\pi}{6}

radians is converted to degrees:

\frac{5\pi}{6} \times \frac{180^\circ}{\pi} = 150^\circ

. Adding these gives the new angle

\theta_Q = -30^\circ + 150^\circ = 120^\circ

. Finally, compute the coordinates using

x = R\cos(120^\circ)

and

y = R\sin(120^\circ)

.

Estimated Time:3m 0s

Question 9Question →

Two connected gears, Gear A and Gear B, rotate together such that the belt connecting them does not slip. The radius of Gear A is $15$ centimeters and the radius of Gear B is $9$ centimeters. If Gear A rotates through a central angle of $\frac{4\pi}{15}$ radians, Gear B rotates through a central angle of $x$ degrees. What is the value of $x$ ?

Show answer & explanation

Answer: 80

Answer

80

The arc length

s

that a point on the belt travels is given by the product of the radius and the angle in radians of Gear A:

s = 15 \times \frac{4\pi}{15} = 4\pi

cm. Since the belt does not slip, Gear B rotates through the same arc length. The angle of Gear B in radians is

\theta = \frac{4\pi}{9}

radians. To convert this angle to degrees, multiply by

\frac{180}{\pi}

to get

\frac{4\pi}{9} \times \frac{180}{\pi} = 80

degrees.

Step-by-Step Solution

1

Calculate the arc length of the rotation for Gear A using the formula

s = r\theta

.

s = 4\pi

centimeters

To find the distance a point on the belt travels, which is shared by both gears.

2

Determine the rotation angle of Gear B in radians using the arc length and Gear B's radius.

\theta_B = \frac{4\pi}{9}

radians

Because the belt does not slip, Gear B must rotate by the same linear arc length as Gear A.

3

Convert the angle of Gear B from radians to degrees by multiplying by

\frac{180}{\pi}

.

x = 80

To find the measure of the angle in degrees as requested by the question.

Key Concept

Converting central angles between radians and degrees in the context of arc lengths of connected circles.

Question 10Question →

The measure of angle $A$ is $45^\circ$ greater than the measure of angle $B$ . If the measure of angle $B$ is $\frac{5\pi}{12}$ radians, what is the measure of angle $A$ , in degrees?

Show answer & explanation

Answer: 120

Answer

120

The correct answer is 120. First, convert the measure of angle

B

from radians to degrees:

\frac{5\pi}{12} \times \frac{180}{\pi} = \frac{5 \times 180}{12} = 5 \times 15 = 75^\circ

. Since angle

A

is

45^\circ

greater than angle

B

, add

45^\circ

to the measure of angle

B

:

75^\circ + 45^\circ = 120^\circ

.

Step-by-Step Solution

1

Convert the measure of angle

B

from radians to degrees.

Angle

B

has a measure of

75^\circ

.

To convert from radians to degrees, multiply the radian measure by

\frac{180}{\pi}

.

2

Calculate the measure of angle

A

by adding

45^\circ

to the measure of angle

B

.

Angle

A

has a measure of

120^\circ

.

It is given that the measure of angle

A

is

45^\circ

greater than the measure of angle

B

.

Key Concept

To convert an angle from radians to degrees, multiply the angle measure in radians by

\frac{180}{\pi}

.

Estimated Time:1m 0s

Question 11Question →

A rotating beacon rotates at a constant rate of $160^\circ$ per second. Through how many radians does the beacon rotate in $4.5$ seconds?

\frac{4}{\pi}

2\pi

4\pi

\frac{32\pi}{9}

Show answer & explanation

Answer: $4\pi$

Answer

The correct answer is the value of

4\pi

radians.

The total angle of rotation in degrees is found by multiplying the rate of rotation by the time elapsed:

160^\circ/\text{second} \times 4.5\text{ seconds} = 720^\circ

. To convert this angle from degrees to radians, multiply the degree measure by the conversion factor

\frac{\pi}{180^\circ}

. This yields

720 \times \frac{\pi}{180} = 4\pi

radians.

Step-by-Step Solution

1

Calculate the total angle of rotation in degrees by multiplying the rate of rotation by the time.

160^\circ \times 4.5 = 720^\circ

To find the total angular displacement, we multiply the constant angular speed by the duration of the rotation.

2

Convert the total angle from degrees to radians by multiplying by the conversion factor

\frac{\pi}{180^\circ}

.

720^\circ \times \frac{\pi}{180^\circ} = 4\pi

radians

Since

180^\circ

is equivalent to

\pi

radians, multiplying by

\frac{\pi}{180^\circ}

converts the angle to radians.

Key Concept

Radian-degree conversion

Alternative Method

Convert the speed of rotation to radians per second first:

160^\circ \times \frac{\pi}{180^\circ} = \frac{8\pi}{9}

radians per second. Then multiply by the time elapsed:

\frac{8\pi}{9} \times 4.5 = 4\pi

radians.

Estimated Time:1m 0s

Question 12Question →

A pendulum swings through an angle of $40^\circ$ , and the tip of the pendulum travels an arc of length $8\pi$ inches. What is the length of the pendulum, in inches?

Show answer & explanation

Answer: 36

Answer

36

To find the length of the pendulum, which represents the radius

r

of the circular path it sweeps, we can use the arc length formula

s = r\theta

, where

s

is the arc length and

\theta

is the central angle in radians. First, convert the given angle from degrees to radians:

\theta = 40^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{9}

radians. Next, substitute the arc length

s = 8\pi

and the angle

\theta = \frac{2\pi}{9}

into the formula:

8\pi = r \left(\frac{2\pi}{9}\right)

. Solving for

r

by multiplying both sides by

\frac{9}{2\pi}

gives

r = 36

. Alternatively, you can use the ratio of the sector's central angle to the total angle of a circle:

\frac{40^\circ}{360^\circ} = \frac{1}{9}

. This means the arc length is

\frac{1}{9}

of the circumference of the circle:

8\pi = \frac{1}{9}(2\pi r)

. Dividing both sides by

2\pi

yields

4 = \frac{1}{9}r

, so

r = 36

.

Step-by-Step Solution

1

Convert the swing angle of the pendulum from degrees to radians.

\theta = \frac{2\pi}{9}

radians

The arc length formula

s = r\theta

requires the angle

\theta

to be in radians.

2

Set up the arc length equation using

s = r\theta

, where

s = 8\pi

is the arc length and

r

is the length of the pendulum.

8\pi = r \left(\frac{2\pi}{9}\right)

The tip of the pendulum travels along a circular path whose radius is the length of the pendulum.

3

Solve the equation for the radius

r

.

r = 36

Isolating

r

by multiplying both sides by

\frac{9}{2\pi}

yields the length of the pendulum.

Key Concept

Converting angle measures between degrees and radians and applying the arc length formula.

Question 13Question →

Angle $A$ has a measure of $d$ degrees, and angle $B$ has a measure of $r$ radians. The sum of the degree measure of angle $A$ and the degree equivalent of the measure of angle $B$ is $180$ . If the measure of angle $A$ , in degrees, is $3$ times the degree equivalent of the measure of angle $B$ , and $r = k\pi$ , what is the value of $k$ ?

Show answer & explanation

Answer: 0.25

Answer

The value of

k

is

0.25

(or

\frac{1}{4}

).

The correct answer is

0.25

(or

\frac{1}{4}

). To find this, we solve the system of equations representing the degree relationship:

d + D_B = 180

and

d = 3D_B

, where

D_B

is the degree equivalent of angle

B

. This gives

D_B = 45

. Converting

45^\circ

to radians by multiplying by

\frac{\pi}{180}

gives

\frac{\pi}{4}

radians, which means the coefficient

k

is

0.25

.

Step-by-Step Solution

1

Set up a system of equations using the given information.

d + D_B = 180

and

d = 3D_B

, where

d

is the degree measure of angle

A

and

D_B

is the degree equivalent of angle

B

.

To translate the verbal descriptions of the relationships between the angle measures into mathematical equations.

2

Solve the system of equations for

D_B

.

D_B = 45

Substituting

d = 3D_B

into the first equation yields

3D_B + D_B = 180

, which simplifies to

4D_B = 180

. Dividing both sides by

4

gives

D_B = 45

.

3

Convert the degree measure of angle

B

to radians.

r = \frac{\pi}{4}

radians

To convert degrees to radians, multiply the degree measure by

\frac{\pi}{180}

. This gives

45 \times \frac{\pi}{180} = \frac{\pi}{4}

.

4

Find the value of

k

from the expression

r = k\pi

.

k = 0.25

(or

\frac{1}{4}

)

Since

r = \frac{\pi}{4} = 0.25\pi

, comparing this to

r = k\pi

shows that

k = 0.25

.

Key Concept

Converting degrees to radians

Question 14Question →

In the $xy$ -plane, an angle $\theta$ is in standard position. The terminal ray of the angle is rotated counterclockwise by $\frac{7\pi}{6}$ radians, and then rotated clockwise by $45^\circ$ . If the terminal ray of the resulting angle lies on the positive $y$ -axis, which of the following could be the value of $\theta$ , in degrees?

15^\circ

195^\circ

255^\circ

285^\circ

Show answer & explanation

Answer: $285^\circ$

Answer

The value of

\theta

could be

285^\circ

.

To find the value of

\theta

, we first convert the counterclockwise rotation of

\frac{7\pi}{6}

radians into degrees. Multiplying

\frac{7\pi}{6}

by

\frac{180}{\pi}

yields

210^\circ

. A counterclockwise rotation increases the angle measure, so we add

210^\circ

. A clockwise rotation decreases the angle measure, so we subtract

45^\circ

. The resulting angle is

\theta + 210^\circ - 45^\circ = \theta + 165^\circ

. Since the terminal ray of the resulting angle lies on the positive

y

-axis, it is coterminal with

90^\circ

. Setting

\theta + 165^\circ = 90^\circ + 360^\circ k

for an integer

k

gives

\theta = -75^\circ + 360^\circ k

. For

k = 1

, we get

\theta = 285^\circ

, which is the correct value.

Step-by-Step Solution

1

Convert the counterclockwise rotation from radians to degrees.

\frac{7\pi}{6} \times \frac{180}{\pi} = 210^\circ

To work with degrees consistently, we convert the radian measure of the rotation using the conversion factor

\frac{180}{\pi}

.

2

Set up the equation for the final angle position using the directions of rotation.

Final angle =

\theta + 210^\circ - 45^\circ = \theta + 165^\circ

Counterclockwise rotations are positive (added to

\theta

) and clockwise rotations are negative (subtracted from the result).

3

Relate the final angle to the positive

y

-axis and solve for

\theta

.

\theta + 165^\circ = 90^\circ + 360^\circ k \implies \theta = -75^\circ + 360^\circ k

. For

k = 1

,

\theta = 285^\circ

.

The positive

y

-axis corresponds to

90^\circ

in standard position. Solving for

\theta

and adding multiples of

360^\circ

yields the possible measures of the starting angle.

Key Concept

Converting angle measures between radians and degrees, determining the direction of angular rotation, and finding coterminal angles.

Question 15Question →

In a circle with center $O$ , the central angle $AOB$ has a measure of $\frac{4\pi}{5}$ radians. Angle $BOC$ is adjacent to angle $AOB$ such that they form a straight line. What is the degree measure of angle $BOC$ ?

Show answer & explanation

Answer: 36

Answer

The degree measure of angle

BOC

is 36.

Angles

AOB

and

BOC

form a straight line, which means they are supplementary and their measures add up to

180^\circ

(or

\pi

radians). Since angle

AOB

measures

\frac{4\pi}{5}

radians, we can convert this measure to degrees first by multiplying by

\frac{180^\circ}{\pi}

:

\frac{4\pi}{5} \times \frac{180^\circ}{\pi} = 4 \times 36^\circ = 144^\circ

. To find the degree measure of the supplementary angle

BOC

, subtract

144^\circ

from

180^\circ

, giving

180^\circ - 144^\circ = 36^\circ

. Alternatively, the calculation can be performed in radians first:

\pi - \frac{4\pi}{5} = \frac{\pi}{5}

radians, and then converting

\frac{\pi}{5}

radians to degrees:

\frac{\pi}{5} \times \frac{180^\circ}{\pi} = 36^\circ

.

Step-by-Step Solution

1

Identify the relationship between the two adjacent angles.

The sum of the measures of angles

AOB

and

BOC

is

\pi

radians or

180^\circ

.

Since the adjacent angles

AOB

and

BOC

form a straight line, they are supplementary.

2

Calculate the measure of angle

BOC

in radians.

Angle

BOC

measures

\frac{\pi}{5}

radians.

Subtract the measure of angle

AOB

from the straight line measure:

\pi - \frac{4\pi}{5} = \frac{\pi}{5}

radians.

3

Convert the radian measure of angle

BOC

to degrees.

The degree measure is 36.

Multiply the radian measure by the conversion factor

\frac{180^\circ}{\pi}

:

\frac{\pi}{5} \times \frac{180}{\pi} = 36^\circ

.

Key Concept

Converting angles from radians to degrees and using the properties of supplementary angles.

Question 16Question →

A circular dial on a vintage radio is rotated by $\frac{5\pi}{8}$ radians to tune to a specific station. If the dial is then rotated by an additional $45^\circ$ in the same direction, what is the total angle of rotation, in radians, of the dial?

\frac{3\pi}{4}

\frac{7\pi}{8}

\frac{9\pi}{8}

\frac{\pi}{2}

Show answer & explanation

Answer: $\frac{7\pi}{8}$

Answer

The correct answer is

\frac{7\pi}{8}

radians.

To find the total angle of rotation in radians, the rotation of

45^\circ

must first be converted to radians by multiplying by

\frac{\pi}{180^\circ}

, which yields

\frac{\pi}{4}

radians. Expressed with a common denominator of

8

, this is equivalent to

\frac{2\pi}{8}

radians. Adding this to the initial rotation of

\frac{5\pi}{8}

radians gives a total rotation of

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

Step-by-Step Solution

1

Convert the additional rotation angle of

45^\circ

into radians.

Since

180^\circ = \pi

radians, the conversion is

45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}

radians.

To find the total rotation angle in radians, both individual angles must be in the same unit of measure.

2

Add the first rotation angle of

\frac{5\pi}{8}

radians to the converted second rotation angle of

\frac{\pi}{4}

radians.

First, express

\frac{\pi}{4}

with a common denominator of

8

:

\frac{\pi}{4} = \frac{2\pi}{8}

. Then, add the two fractions:

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

The total angle of rotation is the sum of the two sequential rotations in the same direction.

Key Concept

To find the sum of angles given in different units, convert the angle measured in degrees to radians using the conversion factor

\frac{\pi \text{ radians}}{180^\circ}

, and then find the sum of the two radian values using a common denominator.

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