Unit Circle and Angle Measures (Radians and Degrees) Practice Questions — ACT

Question 1Question →

An angle in standard position measures $\frac{2\pi}{3}$ radians. If the angle is increased by $45^\circ$ , what is the measure of the new angle, in radians?

\frac{3\pi}{7}

\frac{14\pi}{3}

\frac{12\pi}{11}

\frac{11\pi}{12}

\frac{3\pi}{4}

Show answer & explanation

Answer: $\frac{11\pi}{12}$

Answer

The correct answer is

\frac{11\pi}{12}

To find the final angle measure, first convert the rotation angle of

45^\circ

into radians. Since

180^\circ = \pi

radians, multiplying

45^\circ

by

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

gives

\frac{\pi}{4}

radians. Next, add the initial angle and the rotation:

\frac{2\pi}{3} + \frac{\pi}{4}

. Finding a common denominator of

12

, the sum is

\frac{8\pi}{12} + \frac{3\pi}{12} = \frac{11\pi}{12}

radians.

Step-by-Step Solution

1

Convert the rotation angle from degrees to radians.

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

radians

To perform the addition, both angle measures must be in the same unit (radians).

2

Add the two radian measures.

\frac{2\pi}{3} + \frac{\pi}{4} = \frac{8\pi}{12} + \frac{3\pi}{12} = \frac{11\pi}{12}

radians

An increase in angle measure corresponds to counterclockwise rotation, which means adding the two angles.

Key Concept

Converting between degrees and radians and adding angles in standard position.

Estimated Time:45s

Question 2Question →

Match each angle measure in degrees on the left to its equivalent angle measure in radians on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The degree measures

30^\circ

,

45^\circ

,

60^\circ

, and

90^\circ

correspond to

\frac{\pi}{6}

,

\frac{\pi}{4}

,

\frac{\pi}{3}

, and

\frac{\pi}{2}

radians, respectively.

Each degree measure matches its correct radian value by multiplying the degree measure by

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

and simplifying the fraction.

Step-by-Step Solution

1

Apply the degree-to-radian conversion formula.

Multiply each degree measure by the conversion factor

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

.

A full circle has

360^\circ

or

2\pi

radians, meaning

180^\circ = \pi

radians. Therefore, the conversion factor from degrees to radians is

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

.

2

Simplify the resulting fractions.

30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6}

,

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

,

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

, and

90^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{2}

.

Reducing the fractions by dividing the numerator and denominator by their greatest common factor gives the simplified radian values.

Key Concept

Converting degree measures to radian measures on the unit circle

Estimated Time:45s

Question 3Question →

A central angle of a circle measures $315^\circ$ . What is the radian measure of this angle?

\frac{4}{7\pi}

\frac{4\pi}{7}

\frac{2\pi}{3}

\frac{7\pi}{8}

\frac{7\pi}{4}

Show answer & explanation

Answer: $\frac{7\pi}{4}$

Answer

\frac{7\pi}{4}

To convert degrees to radians, multiply the degree measure by

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

. Multiplying

315^\circ

by

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

gives

\frac{315\pi}{180}

. Simplifying the fraction by dividing the numerator and the denominator by their greatest common divisor, 45, results in

\frac{7\pi}{4}

radians.

Step-by-Step Solution

1

Set up the conversion from degrees to radians.

Multiply

315^\circ

by the conversion factor

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

.

The conversion factor from degrees to radians is

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

because a straight angle of

180^\circ

is equivalent to

\pi

radians.

2

Perform the multiplication and simplify the resulting fraction.

315 \times \frac{\pi}{180} = \frac{315\pi}{180} = \frac{7\pi}{4}

.

Dividing both the numerator 315 and the denominator 180 by their greatest common divisor, 45, yields the simplified fraction

\frac{7}{4}

.

Key Concept

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

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

and simplify the resulting fraction.

Estimated Time:45s

Question 4Question →

An angle in standard position has a measure of $\frac{5\pi}{6}$ radians. What is the degree measure of this angle?

Show answer & explanation

Answer: 150

Answer

The degree measure of the angle is 150.

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

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

. In this case, multiplying

\frac{5\pi}{6}

by

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

simplifies to

150^\circ

because the

\pi

terms cancel and

180

divided by

6

is

30

, which is then multiplied by

5

.

Step-by-Step Solution

1

Multiply the given radian measure by the conversion factor

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

to convert from radians to degrees.

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

One full rotation is

360^\circ

, which is equal to

2\pi

radians. Therefore,

180^\circ = \pi

radians, yielding the conversion factor

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

.

2

Simplify the expression by canceling out common terms.

150^\circ

The common term

\pi

cancels out from the numerator and denominator, leaving

\frac{5 \times 180^\circ}{6} = 5 \times 30^\circ = 150^\circ

.

Key Concept

Converting radian measures to degree measures

Question 5Question →

On the unit circle, a terminal ray starts at the positive $x$ -axis and rotates counterclockwise by $120^\circ$ , and then rotates counterclockwise by an additional $135^\circ$ . What is the radian measure of the final angle in standard position?

\frac{5\pi}{12}

\frac{5\pi}{7}

\frac{17\pi}{12}

\frac{17\pi}{24}

\frac{45900}{\pi}

Show answer & explanation

Answer: $\frac{17\pi}{12}$

Answer

\frac{17\pi}{12}

The correct option is

\frac{17\pi}{12}

. First, the two counterclockwise rotations are added to find the total angle in standard position:

120^\circ + 135^\circ = 255^\circ

. To convert this angle into radians, it is multiplied by the conversion ratio

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

, giving

\frac{255\pi}{180}

. Dividing the numerator and denominator by 15 simplifies the fraction to

\frac{17\pi}{12}

.

Step-by-Step Solution

1

Calculate the total counterclockwise rotation angle in degrees by summing the two individual angles.

120^\circ + 135^\circ = 255^\circ

Consecutive rotations in the same direction add together to find the final angle measure.

2

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

255^\circ \times \frac{\pi}{180^\circ} = \frac{255\pi}{180}

Since 180 degrees equals pi radians, multiplying by pi over 180 converts degrees to radians.

3

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

\frac{255\pi}{180} = \frac{17\pi}{12}

Simplifying the fraction by dividing both 255 and 180 by 15 yields the simplest form.

Key Concept

Unit Circle and Angle Measures (Radians and Degrees)

Question 6Question →

For an angle $\theta$ in standard position, match each description of its terminal side on the left with the corresponding coordinates of its intersection point on the unit circle on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The terminal side in Quadrant II with reference angle

60^\circ

matches

(-\frac{1}{2}, \frac{\sqrt{3}}{2})

; in Quadrant III with reference angle

45^\circ

matches

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

; in Quadrant IV with reference angle

30^\circ

matches

(\frac{\sqrt{3}}{2}, -\frac{1}{2})

; in Quadrant III with reference angle

30^\circ

matches

(-\frac{\sqrt{3}}{2}, -\frac{1}{2})

.

Each terminal side is matched correctly to its coordinates by applying the quadrant signs to the trigonometric values of the reference angles. Quadrant II corresponds to

(-, +)

, Quadrant III corresponds to

(-, -)

, and Quadrant IV corresponds to

(+, -)

. Using standard unit circle coordinates, a

60^\circ

reference angle gives magnitudes of

(\frac{1}{2}, \frac{\sqrt{3}}{2})

, a

45^\circ

reference angle gives

(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})

, and a

30^\circ

reference angle gives

(\frac{\sqrt{3}}{2}, \frac{1}{2})

.

Step-by-Step Solution

1

Determine the signs of the

x

- and

y

-coordinates based on the quadrant of the terminal side.

Quadrant II points have

(-, +)

coordinates; Quadrant III points have

(-, -)

coordinates; Quadrant IV points have

(+, -)

coordinates.

On the unit circle,

x = \cos\theta

and

y = \sin\theta

. Cosine is negative in Quadrants II and III, while sine is negative in Quadrants III and IV.

2

Find the absolute values of the coordinates using the reference angle.

A

30^\circ

reference angle corresponds to coordinates of magnitude

(\frac{\sqrt{3}}{2}, \frac{1}{2})

; a

45^\circ

reference angle corresponds to

(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})

; a

60^\circ

reference angle corresponds to

(\frac{1}{2}, \frac{\sqrt{3}}{2})

.

The reference angle determines the basic trigonometric values

\cos\theta_{\text{ref}}

and

\sin\theta_{\text{ref}}

.

3

Combine the quadrant signs and coordinate magnitudes to find the unique point.

Quadrant II with

60^\circ

reference angle is

(-\frac{1}{2}, \frac{\sqrt{3}}{2})

. Quadrant III with

45^\circ

reference angle is

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

. Quadrant IV with

30^\circ

reference angle is

(\frac{\sqrt{3}}{2}, -\frac{1}{2})

. Quadrant III with

30^\circ

reference angle is

(-\frac{\sqrt{3}}{2}, -\frac{1}{2})

.

Applying the quadrant signs from Step 1 to the magnitude values from Step 2 yields the exact coordinates on the unit circle.

Key Concept

Coordinates of points on the unit circle are given by

(\cos\theta, \sin\theta)

, where the magnitude is determined by the reference angle and the signs are determined by the quadrant of the angle.

Question 7Question →

A particle starts at the point $(1, 0)$ on the unit circle in the standard coordinate plane. It first travels along the circle counterclockwise by $\frac{11\pi}{4}$ radians, then travels clockwise by $120^\circ$ , and finally travels counterclockwise by $\frac{\pi}{4}$ radians. Which of the following ordered pairs represents the coordinates of the particle's final position?

\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)

\left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right)

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

(1, 0)

\left( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right)

Show answer & explanation

Answer: $\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$

Answer

\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)

The correct answer is the coordinate pair representing a net rotation of

\frac{\pi}{3}

radians. Since the particle starts at

(1, 0)

, its coordinates after a net rotation of

\frac{\pi}{3}

radians are given by

(\cos(\frac{\pi}{3}), \sin(\frac{\pi}{3}))

, which evaluates to

\left( \frac{1}{2}, \frac{\sqrt{3}}{2} \right)

.

Step-by-Step Solution

1

Determine the sign and radian value of each rotation.

The first rotation (counterclockwise) is

+\frac{11\pi}{4}

radians. The second rotation (clockwise) is negative:

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

radians. The third rotation (counterclockwise) is

+\frac{\pi}{4}

radians.

Standard orientation defines counterclockwise rotation as positive and clockwise rotation as negative. Angles must be in the same unit (radians) to be combined.

2

Calculate the net angle of rotation by summing the signed values.

\theta_{net} = \frac{11\pi}{4} - \frac{2\pi}{3} + \frac{\pi}{4} = \left(\frac{11\pi}{4} + \frac{\pi}{4}\right) - \frac{2\pi}{3} = 3\pi - \frac{2\pi}{3} = \frac{7\pi}{3}

radians.

Grouping terms with the same denominator simplifies the fraction arithmetic.

3

Find the coterminal angle of

\frac{7\pi}{3}

in the interval

[0, 2\pi)

.

\frac{7\pi}{3} = 2\pi + \frac{\pi}{3}

, which is coterminal to

\frac{\pi}{3}

radians.

Subtracting integer multiples of

2\pi

(full revolutions) gives the standard position of the terminal ray.

4

Evaluate the coordinates of the point on the unit circle at

\frac{\pi}{3}

radians.

(x, y) = \left(\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)

.

For any angle

\theta

on the unit circle, the coordinates are given by

(\cos\theta, \sin\theta)

.

Key Concept

Unit circle coordinates, angle conversion, and coterminal angles

Estimated Time:2m 0s

Question 8Question →

For an angle in standard position on the unit circle, match each rotation scenario on the left with its corresponding terminal angle and location on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Matching the scenarios: the wheel rotation matches

\frac{3\pi}{2}

radians on the negative

y

-axis; the combined rotation of

\frac{5\pi}{3}

and

-450^\circ

matches

\frac{7\pi}{6}

radians in Quadrant III; the clockwise travel of

\frac{11\pi}{4}

units matches

\frac{5\pi}{4}

radians in Quadrant III; the combined rotation of

-150^\circ

and

\frac{11\pi}{6}

matches

\pi

radians on the negative

x

-axis.

Each scenario is correctly matched by converting all angular values to radians, determining the net rotation direction (positive for counterclockwise, negative for clockwise), and finding the coterminal angle in the range

[0, 2\pi)

to locate the terminal side.

Step-by-Step Solution

1

Convert the rotation from revolutions to radians for the first scenario.

3.75 \text{ revolutions} \times 2\pi \text{ radians/revolution} = 7.5\pi

radians. Subtract

3

full rotations (

6\pi

radians) to find the coterminal angle in

[0, 2\pi)

:

7.5\pi - 6\pi = 1.5\pi = \frac{3\pi}{2}

radians. This lies on the negative

y

-axis.

One full revolution corresponds to

2\pi

radians, and subtracting multiples of

2\pi

yields the coterminal position.

2

Calculate the net angle in radians for the second scenario.

Convert

450^\circ

to radians:

-450^\circ \times \frac{\pi}{180^\circ} = -\frac{5\pi}{2}

radians (negative due to clockwise direction). Net angle is

\frac{5\pi}{3} - \frac{5\pi}{2} = -\frac{5\pi}{6}

radians. Find the positive coterminal angle:

-\frac{5\pi}{6} + 2\pi = \frac{7\pi}{6}

radians. This lies in Quadrant III.

Converting all angles to radians with correct sign conventions allows addition to find the net angle.

3

Relate arc length to angle measure on the unit circle for the third scenario.

On a circle with

r = 1

, the arc length

s = \frac{11\pi}{4}

corresponds to a rotation of

\frac{11\pi}{4}

radians. Clockwise direction makes it

-\frac{11\pi}{4}

radians. Find the coterminal angle in

[0, 2\pi)

:

-\frac{11\pi}{4} + 4\pi = \frac{5\pi}{4}

radians. This lies in Quadrant III.

The arc length formula

s = r\theta

simplifies to

s = \theta

on the unit circle, and clockwise motion represents a negative angle.

4

Compute the net angle in radians for the fourth scenario.

Convert

-150^\circ

to radians:

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

radians. Net angle is

-\frac{5\pi}{6} + \frac{11\pi}{6} = \frac{6\pi}{6} = \pi

radians. This lies on the negative

x

-axis.

Converting degrees to radians enables direct fraction addition to determine the final terminal position.

Key Concept

Calculating coterminal angles and conversions between degrees, radians, and revolutions on the unit circle.

Estimated Time:2m 30s

Question 9Question →

An object on a unit circle starts at the point $(1,0)$ and rotates counterclockwise through an angle of $990^\circ$ . It then rotates clockwise through an angle of $\frac{13\pi}{4}$ radians. At which of the following coordinate points on the unit circle does the object finish its path?

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

(0, -1)

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

(-1, 0)

(0, 1)

Show answer & explanation

Answer: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

Answer

The correct coordinate point on the unit circle is

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

.

The correct coordinate point is found by converting the counterclockwise rotation of

990^\circ

to

\frac{11\pi}{2}

radians. Subtracting the clockwise rotation of

\frac{13\pi}{4}

radians yields

\frac{9\pi}{4}

radians. Simplifying this to its coterminal angle in the interval

[0, 2\pi)

gives

\frac{\pi}{4}

radians. The coordinates of

\frac{\pi}{4}

on the unit circle are

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

.

Step-by-Step Solution

1

Convert the initial counterclockwise rotation from degrees to radians.

990^\circ \times \frac{\pi}{180^\circ} = \frac{11\pi}{2}

radians.

To perform calculations with the second rotation which is given in radians, both angle measures should be in the same unit.

2

Subtract the clockwise rotation of

\frac{13\pi}{4}

radians from the first rotation.

\frac{11\pi}{2} - \frac{13\pi}{4} = \frac{22\pi}{4} - \frac{13\pi}{4} = \frac{9\pi}{4}

radians.

Clockwise rotation corresponds to subtracting the angle from the current position.

3

Find the coterminal angle of

\frac{9\pi}{4}

radians within the standard interval

[0, 2\pi)

.

\frac{9\pi}{4} - 2\pi = \frac{\pi}{4}

radians.

Subtracting full rotations (

2\pi

radians) simplifies the angle to its principal equivalent on the unit circle.

4

Determine the coordinates on the unit circle corresponding to the angle

\frac{\pi}{4}

.

\left(\cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right)\right) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)

.

On the unit circle, the coordinates of a point at angle

\theta

are

(\cos(\theta), \sin(\theta))

.

Key Concept

Finding the position on the unit circle after multiple rotations by converting degrees to radians, calculating the coterminal angle, and evaluating coordinate values.

Estimated Time:2m 0s

Question 10Question →

An angle $\theta$ in standard position is coterminal with an angle of $-\frac{17\pi}{4}$ radians. If $0 \le \theta < 2\pi$ , what is the value of $\theta$ expressed as a decimal multiple of $\pi$ ? (For example, if the angle were $\frac{3\pi}{2} = 1.5\pi$ , the answer would be 1.5.)

Show answer & explanation

Answer: 1.75

Answer

1.75

To find a coterminal angle in the interval

[0, 2\pi)

for

-\frac{17\pi}{4}

, we add multiples of

2\pi

. Since

-\frac{17\pi}{4} = -4.25\pi

, we add

6\pi

(three full rotations) to get

-4.25\pi + 6\pi = 1.75\pi

. The multiple of

\pi

is therefore 1.75.

Step-by-Step Solution

1

Convert the coefficient of the given angle from a fraction to a decimal.

-\frac{17}{4} = -4.25

Converting the fraction to a decimal makes it easier to work with the addition of full rotations.

2

Add multiples of

2\pi

(which corresponds to adding 2 to the coefficient of

\pi

) to find a coterminal angle in the interval

[0, 2\pi)

.

-4.25 + 2 = -2.25

;

-2.25 + 2 = -0.25

;

-0.25 + 2 = 1.75

Adding

2\pi

representing full counterclockwise rotations on the unit circle results in a coterminal angle. We repeat this process until the coefficient lies in the interval

[0, 2)

.

3

Identify the coefficient of

\pi

for the coterminal angle.

1.75

The question asks for the angle as a decimal multiple of

\pi

, which is the coefficient of

\pi

in the expression

1.75\pi

.

Key Concept

Coterminal angles are angles in standard position that share the same terminal side. They can be found by adding or subtracting multiples of

2\pi

radians.

Estimated Time:1m 30s

Question 11Question →

An angle in standard position measures $\theta = \frac{7\pi}{6}$ radians. If the terminal side of the angle is rotated counterclockwise by $120^\circ$ , what is the radian measure of the final angle?

\frac{3\pi}{2}

\pi

\frac{\pi}{2}

\frac{11\pi}{6}

\frac{8\pi}{3}

Show answer & explanation

Answer: $\frac{11\pi}{6}$

Answer

The final angle measures

\frac{11\pi}{6}

radians.

The correct answer is

\frac{11\pi}{6}

radians. To find the final angle, we convert the rotation angle of

120^\circ

into radians:

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

radians. Since the rotation is counterclockwise, we add this value to the initial angle of

\frac{7\pi}{6}

radians:

\frac{7\pi}{6} + \frac{2\pi}{3} = \frac{7\pi}{6} + \frac{4\pi}{6} = \frac{11\pi}{6}

radians.

Step-by-Step Solution

1

Convert the rotation angle from degrees to radians.

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

radians.

Both angles must be in radians to add them directly.

2

Set up the sum for a counterclockwise rotation.

\frac{7\pi}{6} + \frac{2\pi}{3}

Counterclockwise rotation corresponds to adding the angle measure.

3

Find the common denominator and calculate the sum.

\frac{7\pi}{6} + \frac{4\pi}{6} = \frac{11\pi}{6}

radians.

The least common denominator of

6

and

3

is

6

, so we multiply the numerator and denominator of

\frac{2\pi}{3}

by

2

before adding.

Key Concept

Unit Circle and Angle Measures (Radians and Degrees)

Estimated Time:1m 30s

Question 12Question →

For each angle or terminal ray described on the left, match it to the correct coordinates $(x, y)$ of its intersection with the unit circle in the standard coordinate plane on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches associate the angle of 7pi/6 radians with coordinates (-sqrt(3)/2, -1/2); the angle of 5pi/4 radians with coordinates (-sqrt(2)/2, -sqrt(2)/2); the angle of pi/3 radians with coordinates (1/2, sqrt(3)/2); and the y-axis reflected angle of 4pi/3 radians with coordinates (1/2, -sqrt(3)/2).

Each description on the left simplifies to a unique angle in standard position on the unit circle. The coordinates on the right represent the exact

(\cos\theta, \sin\theta)

values for each corresponding angle: the angle of

\frac{7\pi}{6}

radians corresponds to coordinates

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

; the angle of

\frac{5\pi}{4}

radians corresponds to coordinates

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

; the angle of

\frac{\pi}{3}

radians corresponds to coordinates

\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)

; and the terminal ray symmetric to the terminal ray of

\frac{4\pi}{3}

radians across the

y

-axis corresponds to coordinates

\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

.

Step-by-Step Solution

1

Analyze the first angle description by converting all angle measures to a common unit (radians) and computing the net rotation.

For the first angle,

570^\circ = \frac{19\pi}{6}

radians, and the net rotation is

\frac{13\pi}{3} - \frac{19\pi}{6} = \frac{7\pi}{6}

radians.

Converting degrees to radians and subtracting the clockwise rotation from the counterclockwise rotation simplifies the angle to a standard radian value.

2

For the second angle, express the starting point as an angle in radians and add the given counterclockwise rotation, then find the coterminal angle in

[0, 2\pi)

.

Starting at

(0, -1)

is equivalent to

\frac{3\pi}{2}

radians. Total rotation is

\frac{3\pi}{2} + \frac{23\pi}{4} = \frac{29\pi}{4}

radians, which is coterminal with

\frac{29\pi}{4} - 6\pi = \frac{5\pi}{4}

radians.

Using the standard position angle of the starting point allows us to sum the rotations and determine the final terminal ray position.

3

Determine the coterminal angle for

-1020^\circ

in the interval

[0, 360^\circ)

and convert it to radians.

-1020^\circ + 1080^\circ = 60^\circ

, which equals

\frac{\pi}{3}

radians.

Adding multiples of

360^\circ

finds the equivalent positive angle within one full revolution.

4

Find the coordinates of the terminal ray of

\frac{4\pi}{3}

radians on the unit circle, reflect the point across the

y

-axis, and determine the coordinates of the resulting point.

The coordinate point of

\frac{4\pi}{3}

is

\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

. Reflecting this point across the

y

-axis yields

\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

.

Symmetry across the

y

-axis negates the

x

-coordinate of the point on the unit circle.

5

Match each of the simplified angles to their corresponding standard coordinates

(x, y) = (\cos\theta, \sin\theta)

on the unit circle.

The angle

\frac{7\pi}{6}

matches

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

;

\frac{5\pi}{4}

matches

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

;

\frac{\pi}{3}

matches

\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)

; and the reflected terminal ray matches

\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

.

Evaluating the sine and cosine functions at each angle yields the final coordinates.

Key Concept

Identifying terminal coordinates of angles on the unit circle by converting between degrees and radians, calculating coterminal angles, and applying coordinate symmetries.

Estimated Time:3m 0s

Question 13Question →

An angle in standard position measures $-468^\circ$ . What is the radian measure of the coterminal angle that lies in the interval $[0, 2\pi)$ ?

\frac{5\pi}{7}

\frac{7\pi}{5}

\frac{\pi}{6}

\frac{3\pi}{5}

\frac{9\pi}{5}

Show answer & explanation

Answer: $\frac{7\pi}{5}$

Answer

The correct radian measure of the coterminal angle is

\frac{7\pi}{5}

.

To find the correct radian measure, first determine the positive coterminal angle in degrees by adding multiples of

360^\circ

until the angle lies in the interval

[0^\circ, 360^\circ)

. Adding

720^\circ

to

-468^\circ

results in

252^\circ

. Convert this angle to radians by multiplying by

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

, which yields

\frac{252\pi}{180}

. Dividing the numerator and denominator by their greatest common divisor,

36

, simplifies the expression to

\frac{7\pi}{5}

radians.

Step-by-Step Solution

1

Find a positive coterminal angle in degrees by adding multiples of

360^\circ

to the initial angle.

-468^\circ + 360^\circ = -108^\circ

, and

-108^\circ + 360^\circ = 252^\circ

.

Adding

720^\circ

(two full rotations) to

-468^\circ

shifts the angle into the standard positive range

[0^\circ, 360^\circ)

while maintaining the same terminal ray.

2

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

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

.

252^\circ \times \frac{\pi}{180^\circ} = \frac{252\pi}{180}

radians.

Since

\pi

radians corresponds to

180^\circ

, multiplying by

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

changes the unit of measure from degrees to radians.

3

Simplify the fraction

\frac{252\pi}{180}

by dividing both the numerator and the denominator by their greatest common divisor.

Dividing

252

and

180

by their greatest common divisor of

36

yields

\frac{7\pi}{5}

radians.

Simplifying the fraction expresses the final radian measure in its standard, reduced form.

Key Concept

Finding positive coterminal angles and converting degree measures to radian measures on the unit circle.

Alternative Method

Convert the initial angle of

-468^\circ

directly to radians first by multiplying by

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

, yielding

-\frac{13\pi}{5}

radians. To find the positive coterminal angle in the interval

[0, 2\pi)

, add multiples of

2\pi

radians (which is

\frac{10\pi}{5}

):

-\frac{13\pi}{5} + \frac{10\pi}{5} = -\frac{3\pi}{5}

, and then

-\frac{3\pi}{5} + \frac{10\pi}{5} = \frac{7\pi}{5}

radians.

Estimated Time:2m 0s

Question 14Question →

A terminal ray of an angle $\theta$ in standard position passes through the point $\left(-\frac{1}{2}, y\right)$ on the unit circle in Quadrant III. What is the value of $\theta$ in radians, where $0 \le \theta < 2\pi$ ?

\frac{2\pi}{3}

\frac{4\pi}{3}

\frac{5\pi}{4}

\frac{5\pi}{3}

\frac{7\pi}{6}

Show answer & explanation

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

Answer

The correct answer is the option containing the radian measure

\frac{4\pi}{3}

.

The correct answer is the option containing the radian measure

\frac{4\pi}{3}

. Any point

(x, y)

on the unit circle satisfies

x^2 + y^2 = 1

. Substituting

x = -\frac{1}{2}

gives

\frac{1}{4} + y^2 = 1

, which simplifies to

y^2 = \frac{3}{4}

, so

y = \pm \frac{\sqrt{3}}{2}

. Since the terminal ray lies in Quadrant III, the y-coordinate must be negative, so

y = -\frac{\sqrt{3}}{2}

. The angle

\theta

in

[0, 2\pi)

with

\cos(\theta) = -\frac{1}{2}

and

\sin(\theta) = -\frac{\sqrt{3}}{2}

is

\theta = \frac{4\pi}{3}

radians.

Step-by-Step Solution

1

Use the equation of the unit circle,

x^2 + y^2 = 1

, to find the y-coordinate of the point.

Since

x = -\frac{1}{2}

, we have

\left(-\frac{1}{2}\right)^2 + y^2 = 1 \Rightarrow \frac{1}{4} + y^2 = 1 \Rightarrow y^2 = \frac{3}{4}

.

The coordinates of any point on the unit circle must satisfy the equation

x^2 + y^2 = 1

.

2

Determine the correct sign of the y-coordinate using the quadrant information.

y = -\frac{\sqrt{3}}{2}

Since the point lies in Quadrant III, both the x and y coordinates must be negative.

3

Identify the angle

\theta

in the interval

[0, 2\pi)

that corresponds to these coordinates on the unit circle.

\theta = \frac{4\pi}{3}

radians

On the unit circle,

\cos(\theta) = x = -\frac{1}{2}

and

\sin(\theta) = y = -\frac{\sqrt{3}}{2}

. In Quadrant III, the angle with a reference angle of

\frac{\pi}{3}

is

\pi + \frac{\pi}{3} = \frac{4\pi}{3}

.

Key Concept

Finding an angle in standard position on the unit circle given its x-coordinate and quadrant

Alternative Method

Alternatively, one can convert the options from radians to degrees and evaluate their cosine values. The correct angle must satisfy

\cos(\theta) = -1/2

. Since

\cos(2\pi/3) = -1/2

and

\cos(4\pi/3) = -1/2

, we only need to check these two options. Among these, only

\frac{4\pi}{3}

lies in Quadrant III, which confirms it as the correct answer.

Estimated Time:1m 30s

Question 15Question →

An angle $\theta$ in standard position is rotated counterclockwise by $225^\circ$ . The terminal ray of the resulting angle lies in the fourth quadrant along the line $y = -x\sqrt{3}$ . If the radian measure of the smallest positive angle $\theta$ is written in simplest form as $\frac{a\pi}{b}$ , where $a$ and $b$ are positive integers, what is the value of $a + b$ ?

Show answer & explanation

Answer: 17

Answer

The value of

a + b

is

17

.

The resulting angle

\alpha

lies along the line

y = -x\sqrt{3}

in the fourth quadrant, meaning its measure is

300^\circ

plus any multiple of

360^\circ

. Subtracting the counterclockwise rotation of

225^\circ

gives the original angle

\theta = 75^\circ

(for the smallest positive angle). Converting

75^\circ

to radians by multiplying by

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

yields

\frac{5\pi}{12}

. Since the fraction is in simplest form,

a = 5

and

b = 12

, and their sum is

17

.

Step-by-Step Solution

1

Find the angle of the terminal ray after rotation from its equation and quadrant

The terminal ray after rotation is at an angle of

300^\circ

(or

\frac{5\pi}{3}

radians)

The line

y = -x\sqrt{3}

has a slope of

-\sqrt{3}

, so the angle

\alpha

in Quadrant IV satisfies

\tan\alpha = -\sqrt{3}

, which means

\alpha = 300^\circ

.

2

Set up and solve the equation for the original angle

\theta

before the counterclockwise rotation of

225^\circ

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

Since the angle was rotated counterclockwise by

225^\circ

to reach the final position of

300^\circ

, we have

\theta + 225^\circ = 300^\circ + 360^\circ k

.

3

Determine the smallest positive value of

\theta

\theta = 75^\circ

Setting

k = 0

gives the smallest positive angle of

75^\circ

.

4

Convert the angle

\theta

from degrees to radians

\theta = \frac{5\pi}{12}

radians

To convert degrees to radians, multiply by

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

, yielding

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

.

5

Calculate the sum of the numerator and denominator of the simplified radian fraction

a + b = 17

The fraction

\frac{5}{12}

is in simplest form, so

a = 5

and

b = 12

. The sum is

5 + 12 = 17

.

Key Concept

Converting degree measures to radian measures and finding coterminal angles on the unit circle

Question 16Question →

Match each of the degree measures of angles in standard position on the left with its mathematically equivalent radian measure on the right. Which radian measure corresponds to each degree measure?

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings are:

-135^\circ

matches with

-\frac{3\pi}{4}

radians;

480^\circ

matches with

\frac{8\pi}{3}

radians;

-300^\circ

matches with

-\frac{5\pi}{3}

radians; and

585^\circ

matches with

\frac{13\pi}{4}

radians.

Each degree measure is multiplied by

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

and simplified to find its equivalent radian measure. This process yields the unique matching pairs:

-135^\circ

to

-\frac{3\pi}{4}

radians,

480^\circ

to

\frac{8\pi}{3}

radians,

-300^\circ

to

-\frac{5\pi}{3}

radians, and

585^\circ

to

\frac{13\pi}{4}

radians.

Step-by-Step Solution

1

Recall the formula to convert degrees to radians.

Radian measure = Degree measure

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

Since a full circle is

360^\circ

or

2\pi

radians, the conversion ratio simplifies to

\pi

radians per

180^\circ

.

2

Convert the first degree measure,

-135^\circ

, to radians.

-135^\circ \times \frac{\pi}{180^\circ} = -\frac{135\pi}{180} = -\frac{3\pi}{4}

radians

Dividing the numerator and denominator by their greatest common divisor,

45

, simplifies the fraction to

-\frac{3}{4}

.

3

Convert the second degree measure,

480^\circ

, to radians.

480^\circ \times \frac{\pi}{180^\circ} = \frac{480\pi}{180} = \frac{8\pi}{3}

radians

Dividing the numerator and denominator by their greatest common divisor,

60

, simplifies the fraction to

\frac{8}{3}

.

4

Convert the third degree measure,

-300^\circ

, to radians.

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

radians

Dividing the numerator and denominator by their greatest common divisor,

60

, simplifies the fraction to

-\frac{5}{3}

.

5

Convert the fourth degree measure,

585^\circ

, to radians.

585^\circ \times \frac{\pi}{180^\circ} = \frac{585\pi}{180} = \frac{13\pi}{4}

radians

Dividing the numerator and denominator by their greatest common divisor,

45

, simplifies the fraction to

\frac{13}{4}

.

Key Concept

Converting degree measures to equivalent radian measures using the conversion factor

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

.

Alternative Method

Alternatively, you can recall key benchmark angles on the unit circle (such as

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

radians and

60^\circ = \frac{\pi}{3}

radians) and express each angle as an integer multiple of these benchmarks. For example,

-135^\circ

is

-3 \times 45^\circ

, which corresponds to

-3 \times \frac{\pi}{4} = -\frac{3\pi}{4}

radians.

Estimated Time:1m 30s

Question 17Question →

A point on the unit circle starts at $(1, 0)$ and undergoes a counterclockwise rotation of $570^\circ$ about the origin. What is the radian measure of the angle in the interval $[0, 2\pi)$ that corresponds to the point's final position?

\frac{6\pi}{7}

\frac{17\pi}{6}

\frac{7\pi}{6}

\frac{7}{6\pi}

\frac{3\pi}{4}

Show answer & explanation

Answer: $\frac{7\pi}{6}$

Answer

The correct answer is the option representing

\frac{7\pi}{6}

radians.

The correct answer represents

\frac{7\pi}{6}

radians. To find the position on the unit circle after a counterclockwise rotation of

570^\circ

, we first determine the coterminal angle within one full rotation (

360^\circ

). Subtracting

360^\circ

from

570^\circ

gives

210^\circ

. We then convert this angle to radians by multiplying by

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

, which simplifies to

\frac{7\pi}{6}

radians.

Step-by-Step Solution

1

Find a coterminal angle for

570^\circ

within the standard

[0^\circ, 360^\circ)

range.

570^\circ - 360^\circ = 210^\circ

Since a full rotation is

360^\circ

, subtracting

360^\circ

yields an angle in the same position on the unit circle but within one full rotation.

2

Convert the coterminal angle from degrees to radians by multiplying by

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

.

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

radians

To convert degrees to radians, multiply by the conversion factor

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

.

3

Verify that the resulting angle is in the interval

[0, 2\pi)

.

Since

0 \le \frac{7\pi}{6} < 2\pi

, the angle is in the correct interval.

The question requires the final angle to be in the interval

[0, 2\pi)

.

Key Concept

Finding coterminal angles and converting degree measures to radian measures on the unit circle.

Alternative Method

Convert the rotation to radians first:

570^\circ \times \frac{\pi}{180^\circ} = \frac{19\pi}{6}

radians. Then, subtract

2\pi

to find the coterminal angle:

\frac{19\pi}{6} - 2\pi = \frac{7\pi}{6}

radians.

Estimated Time:1m 15s

Question 18Question →

An angle $\theta$ in standard position measures $-750^\circ$ . Let $\phi$ be the coterminal angle of $\theta$ such that $0 \le \phi < 2\pi$ radians. If $\phi = \frac{a\pi}{b}$ , where $a$ and $b$ are positive integers with no common factors, what is the value of $a + b$ ?

Show answer & explanation

Answer: 17

Answer

17

To find the coterminal angle

\phi

in the range

[0, 2\pi)

radians, we first determine the coterminal angle in degrees by adding multiples of

360^\circ

. Adding

3 \times 360^\circ = 1080^\circ

to

-750^\circ

results in

330^\circ

. We then convert

330^\circ

to radians by multiplying by

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

, which simplifies to

\frac{11\pi}{6}

. Since

11

and

6

are positive integers with no common factors,

a = 11

and

b = 6

. The sum

a + b

is

11 + 6 = 17

.

Step-by-Step Solution

1

Find the positive coterminal angle of

-750^\circ

within one full rotation.

330^\circ

Adding

1080^\circ

(three full rotations of

360^\circ

) to

-750^\circ

brings the angle within the standard range of

[0^\circ, 360^\circ)

.

2

Convert the angle from degrees to radians.

\frac{11\pi}{6}

radians

Multiplying the degree measure by

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

and simplifying the fraction converts it to radians.

3

Identify the values of

a

and

b

and compute their sum.

17

Comparing

\frac{11\pi}{6}

to

\frac{a\pi}{b}

shows

a = 11

and

b = 6

, which share no common factors. Their sum is

11 + 6 = 17

.

Key Concept

Finding coterminal angles and converting angle measures between degrees and radians.

Estimated Time:1m 30s

Question 19Question →

A radar signal sweeps counterclockwise around a control tower located at the origin of a coordinate plane. Starting from standard position along the positive $x$ -axis, the radar line rotates through an angle of $\frac{13\pi}{4}$ radians. Which of the following ordered pairs represents the $(x, y)$ coordinates of the point where the radar line intersects the unit circle centered at the origin?

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

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

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

\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

Show answer & explanation

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

Answer

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

Subtracting

2\pi

(one full rotation) from

\frac{13\pi}{4}

gives the coterminal angle

\frac{5\pi}{4}

. This angle terminates in Quadrant III, where both cosine (

x

-coordinate) and sine (

y

-coordinate) are negative. Using the reference angle

\frac{\pi}{4}

, both values have magnitude

\frac{\sqrt{2}}{2}

, giving the coordinate pair

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

.

Step-by-Step Solution

1

Find an equivalent coterminal angle within one full revolution

[0, 2\pi)

.

Subtract

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

from

\frac{13\pi}{4}

:

\frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4}

radians.

Coterminal angles share the exact same terminal ray and unit circle coordinates.

2

Identify the quadrant and reference angle for

\frac{5\pi}{4}

.

The angle lies in Quadrant III because

\pi < \frac{5\pi}{4} < \frac{3\pi}{2}

. The reference angle is

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

.

The reference angle determines the absolute magnitude of the trigonometric coordinates.

3

Calculate the coordinates

(x, y) = (\cos\theta, \sin\theta)

for the terminal ray.

Since

\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

and

\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}

, and both coordinates are negative in Quadrant III,

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

.

Points on the unit circle are defined by

(\cos\theta, \sin\theta)

.

Key Concept

Coterminal Angles and Unit Circle Coordinates

Estimated Time:1m 0s

Question 20Question →

A robotic arm starts at the point $(1,0)$ on the unit circle and rotates counterclockwise by $\frac{11\pi}{6}$ radians. It then rotates clockwise by $120^\circ$ . At which of the following coordinates on the unit circle does the arm's tip end?

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

\left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)

(0, 1)

(-1, 0)

\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)

Show answer & explanation

Answer: $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$

Answer

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

The correct answer is obtained by converting the initial rotation of

\frac{11\pi}{6}

radians to

330^\circ

, subtracting the clockwise rotation of

120^\circ

to get

210^\circ

, and finding the cosine and sine values for this angle in the third quadrant, which are

-\frac{\sqrt{3}}{2}

and

-\frac{1}{2}

respectively.

Step-by-Step Solution

1

Convert the initial counterclockwise rotation from radians to degrees.

\frac{11\pi}{6} \text{ radians} \times \frac{180^\circ}{\pi} = 11 \times 30^\circ = 330^\circ

.

Converting both angles to degrees makes them easier to combine.

2

Apply the second rotation (clockwise, which means subtracting the angle).

330^\circ - 120^\circ = 210^\circ

.

Clockwise rotation reduces the angle in standard position.

3

Find the unit circle coordinates for the final angle of

210^\circ

.

x = \cos(210^\circ) = -\frac{\sqrt{3}}{2}

and

y = \sin(210^\circ) = -\frac{1}{2}

, yielding the coordinates

\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)

.

The terminal ray of

210^\circ

lies in Quadrant III, where both sine and cosine are negative, with a reference angle of

30^\circ

.

Key Concept

To find coordinates of a rotated point on the unit circle, convert the angle measures to a common unit, compute the net rotation angle in standard position, and evaluate the cosine (for the

x

-coordinate) and sine (for the

y

-coordinate) of the resulting angle.

Estimated Time:1m 30s

Unit Circle and Angle Measures (Radians and Degrees)

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