Trigonometry 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 →

For an angle $\theta$ in the interval $\frac{\pi}{2} < \theta < \pi$ , the value of $\cos \theta = -\frac{3}{5}$ . What is the value of $\sin \theta + \cos \theta$ ?

\frac{1}{10}

-\frac{7}{5}

1

-\frac{1}{5}

\frac{1}{5}

Show answer & explanation

Answer: $\frac{1}{5}$

Answer

one-fifth

To evaluate

\sin \theta + \cos \theta

, we first find the value of

\sin \theta

. We substitute

\cos \theta = -\frac{3}{5}

into the Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

, which gives

\sin^2 \theta + \frac{9}{25} = 1

. Solving for

\sin^2 \theta

yields

\frac{16}{25}

. Because the angle

\theta

is constrained to the second quadrant (

\frac{\pi}{2} < \theta < \pi

), its sine value must be positive, which means

\sin \theta = \frac{4}{5}

. Adding the values together, we get

\frac{4}{5} + \left(-\frac{3}{5}\right) = \frac{1}{5}

.

Step-by-Step Solution

1

Use the Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

to find the magnitude of the sine function.

\sin^2 \theta = 1 - \left(-\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}

The Pythagorean identity relates sine and cosine for any angle.

2

Determine the correct sign of

\sin \theta

based on the given quadrant interval.

Since

\frac{\pi}{2} < \theta < \pi

, the angle

\theta

lies in Quadrant II, where the sine function is positive. Thus,

\sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}

.

The trigonometric function values are positive or negative depending on the quadrant on the unit circle.

3

Compute the sum of

\sin \theta

and

\cos \theta

.

\sin \theta + \cos \theta = \frac{4}{5} + \left(-\frac{3}{5}\right) = \frac{1}{5}

This gives the final value of the requested expression.

Key Concept

Using the Pythagorean identity

\sin^2 \theta + \cos^2 \theta = 1

and quadrant rules to calculate trigonometric values.

Alternative Method

We can sketch a reference right triangle in Quadrant II. Since

\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = -\frac{3}{5}

, we assign the adjacent side a length of

-3

along the x-axis and the hypotenuse a length of

5

. By the Pythagorean theorem, the opposite vertical side is

\sqrt{5^2 - (-3)^2} = 4

. Since the vertical side is in Quadrant II, it is positive. This makes

\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5}

. Evaluating the sum yields

\frac{4}{5} + \left(-\frac{3}{5}\right) = \frac{1}{5}

.

Estimated Time:45s

Question 10Question →

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 11Question →

For an angle $\theta$ such that $\pi < \theta < \frac{3\pi}{2}$ , the value of $\cos\theta = -0.8$ . What is the value of $10\sin\theta$ ?

Show answer & explanation

Answer: -6

Answer

The value of

10\sin\theta

is

-6

.

The correct answer is

-6

. Substituting

\cos\theta = -0.8

into the fundamental Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

yields

\sin^2\theta + 0.64 = 1

, which simplifies to

\sin^2\theta = 0.36

. Since the angle

\theta

lies in the interval

\pi < \theta < \frac{3\pi}{2}

(Quadrant III), its sine value must be negative. Thus,

\sin\theta = -0.6

. Multiplying this value by 10 gives

10\sin\theta = -6

.

Step-by-Step Solution

1

Use the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

with the given cosine value

\cos\theta = -0.8

.

\sin^2\theta + (-0.8)^2 = 1 \Rightarrow \sin^2\theta + 0.64 = 1 \Rightarrow \sin^2\theta = 0.36

The Pythagorean identity relates the sine and cosine values of any angle.

2

Take the square root of both sides, selecting the correct sign based on the quadrant constraint

\pi < \theta < \frac{3\pi}{2}

.

Since the angle lies in Quadrant III, the sine function must be negative. Thus,

\sin\theta = -\sqrt{0.36} = -0.6

.

In the third quadrant, y-coordinates on the unit circle are negative, meaning

\sin\theta

must be negative.

3

Multiply

\sin\theta

by 10 to get the final requested value.

10\sin\theta = 10 \times (-0.6) = -6

This calculation yields the final answer requested by the problem.

Key Concept

Applying the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

and choosing the correct sign based on the angle's quadrant.

Question 12Question →

An angle $\theta$ lies in the second quadrant and satisfies $\sin\theta = \frac{5}{13}$ . What is the value of $12\tan\theta$ ?

Show answer & explanation

Answer: -5

Answer

The value of

12\tan\theta

is -5.

Applying the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

with

\sin\theta = \frac{5}{13}

yields

\cos^2\theta = \frac{144}{169}

. Since the angle

\theta

lies in the second quadrant, its cosine is negative, meaning

\cos\theta = -\frac{12}{13}

. Using the quotient identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

, we find

\tan\theta = -\frac{5}{12}

. Multiplying by 12 gives the correct value of -5.

Step-by-Step Solution

1

Use the Pythagorean identity to find the magnitude of

\cos\theta

.

\cos^2\theta = \frac{144}{169}

The Pythagorean identity states that

\sin^2\theta + \cos^2\theta = 1

. Substituting

\sin\theta = \frac{5}{13}

gives

\cos^2\theta = 1 - \left(\frac{5}{13}\right)^2

.

2

Determine the value of

\cos\theta

by applying the quadrant sign rule.

\cos\theta = -\frac{12}{13}

Since

\theta

is in the second quadrant, the cosine of

\theta

must be negative.

3

Calculate

\tan\theta

using the quotient identity.

\tan\theta = -\frac{5}{12}

The quotient identity is

\tan\theta = \frac{\sin\theta}{\cos\theta}

.

4

Multiply

\tan\theta

by 12 to find the required expression's value.

-5

Multiplying the value of

\tan\theta

(which is

-\frac{5}{12}

) by 12 yields

-5

.

Key Concept

Fundamental Trigonometric Identities

Estimated Time:1m 0s

Question 13Question →

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 14Question →

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 15Question →

Given that $\cos\theta = \frac{4}{5}$ and the terminal side of angle $\theta$ lies in Quadrant IV, what is the value of $\tan\theta$ ?

-\frac{4}{3}

-\frac{3}{4}

-\frac{3}{5}

\frac{3}{4}

\frac{4}{3}

Show answer & explanation

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

Answer

-\frac{3}{4}

The correct answer is

-\frac{3}{4}

. Since the angle

\theta

has its terminal side in Quadrant IV, its cosine is positive and its sine is negative. Using the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

, we find

\sin^2\theta + \left(\frac{4}{5}\right)^2 = 1

, which simplifies to

\sin^2\theta = \frac{9}{25}

. Because sine is negative in Quadrant IV,

\sin\theta = -\frac{3}{5}

. Finally, applying the quotient identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

, we get

\tan\theta = \frac{-3/5}{4/5} = -\frac{3}{4}

.

Step-by-Step Solution

1

Determine the sign of

\sin\theta

in Quadrant IV.

\sin\theta < 0

In Quadrant IV, the x-coordinates (representing cosine) are positive, and the y-coordinates (representing sine) are negative.

2

Use the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

to calculate the value of

\sin\theta

.

\sin\theta = -\frac{3}{5}

Substituting

\cos\theta = \frac{4}{5}

gives

\sin^2\theta + \left(\frac{4}{5}\right)^2 = 1 \implies \sin^2\theta = 1 - \frac{16}{25} = \frac{9}{25}

. Taking the negative square root because sine is negative in Quadrant IV yields

-\frac{3}{5}

.

3

Use the quotient identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

to calculate

\tan\theta

.

\tan\theta = -\frac{3}{4}

Dividing the value of

\sin\theta

by

\cos\theta

yields

\frac{-3/5}{4/5} = -\frac{3}{4}

.

Key Concept

Fundamental Trigonometric Identities

Question 16Question →

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 17Question →

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 18Question →

Match each of the trigonometric functions listed on the left with the correct description of its amplitude and period listed on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The function

y = 3\sin(2x)

matches the description stating 'Amplitude is 3 and period is

\pi

'. The function

y = 2\cos(3x)

matches the description stating 'Amplitude is 2 and period is

\frac{2\pi}{3}

'. The function

y = 4\sin(\pi x)

matches the description stating 'Amplitude is 4 and period is 2'.

Each trigonometric function of the form

y = a\sin(bx)

or

y = a\cos(bx)

has an amplitude equal to the absolute value of the coefficient of the trigonometric term (

|a|

) and a period equal to

2\pi

divided by the absolute value of the coefficient of the angle variable (

|b|

). Applying these formulas gives the correct properties for each function.

Step-by-Step Solution

1

Identify the general form of the trigonometric functions

The functions are in the form

y = a\sin(bx)

or

y = a\cos(bx)

, where the amplitude is given by the absolute value of the vertical stretch coefficient (

|a|

), and the period is calculated as

\frac{2\pi}{|b|}

.

This establishes the formulas needed to determine the amplitude and period for each equation.

2

Calculate the properties for

y = 3\sin(2x)

The vertical stretch coefficient is 3, so the amplitude is 3. The frequency coefficient is 2, so the period is

\frac{2\pi}{2} = \pi

.

To find the amplitude and period for the first function.

3

Calculate the properties for

y = 2\cos(3x)

The vertical stretch coefficient is 2, so the amplitude is 2. The frequency coefficient is 3, so the period is

\frac{2\pi}{3}

.

To find the amplitude and period for the second function.

4

Calculate the properties for

y = 4\sin(\pi x)

The vertical stretch coefficient is 4, so the amplitude is 4. The frequency coefficient is

\pi

, so the period is

\frac{2\pi}{\pi} = 2

.

To find the amplitude and period for the third function.

Key Concept

Identifying the amplitude and calculating the period of trigonometric functions from their equations

Estimated Time:1m 30s

Question 19Question →

An angle $\theta$ satisfies $\frac{\pi}{2} < \theta < \pi$ and $(\sin\theta - \cos\theta)^2 = \frac{17}{9}$ . What is the value of $\tan\theta + \cot\theta$ ?

1

-\frac{9}{8}

-\frac{9}{4}

\frac{9}{4}

-\frac{4}{9}

Show answer & explanation

Answer: $-\frac{9}{4}$

Answer

The value of the expression is

-\frac{9}{4}

Expanding the square of the difference

(\sin\theta - \cos\theta)^2

yields

\sin^2\theta - 2\sin\theta\cos\theta + \cos^2\theta

. Applying the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

simplifies this expression to

1 - 2\sin\theta\cos\theta

. Setting this equal to

\frac{17}{9}

and solving for the product of sine and cosine gives

\sin\theta\cos\theta = -\frac{4}{9}

. The target expression

\tan\theta + \cot\theta

can be rewritten using quotient and reciprocal identities as

\frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}

, which simplifies by finding a common denominator to

\frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} = \frac{1}{\sin\theta\cos\theta}

. Substituting the value of

\sin\theta\cos\theta

into this expression results in

-\frac{9}{4}

.

Step-by-Step Solution

1

Expand the squared expression

(\sin\theta - \cos\theta)^2

and apply the Pythagorean identity.

(\sin\theta - \cos\theta)^2 = \sin^2\theta - 2\sin\theta\cos\theta + \cos^2\theta = 1 - 2\sin\theta\cos\theta

To express the squared binomial in terms of the product

\sin\theta\cos\theta

using the identity

\sin^2\theta + \cos^2\theta = 1

.

2

Equate the expanded form to the given value and solve for

\sin\theta\cos\theta

.

1 - 2\sin\theta\cos\theta = \frac{17}{9} \implies 2\sin\theta\cos\theta = -\frac{8}{9} \implies \sin\theta\cos\theta = -\frac{4}{9}

To find the value of the product

\sin\theta\cos\theta

from the given equation.

3

Rewrite the target expression

\tan\theta + \cot\theta

in terms of sine and cosine.

\tan\theta + \cot\theta = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta} = \frac{\sin^2\theta + \cos^2\theta}{\sin\theta\cos\theta} = \frac{1}{\sin\theta\cos\theta}

To simplify the target sum of ratios using quotient and reciprocal identities so that it depends only on the product

\sin\theta\cos\theta

.

4

Substitute the value of

\sin\theta\cos\theta

into the simplified expression.

\tan\theta + \cot\theta = \frac{1}{-4/9} = -\frac{9}{4}

To compute the final numerical value of the expression.

Key Concept

Simplifying trigonometric expressions using fundamental Pythagorean, quotient, and reciprocal identities, and solving for unknown products of trigonometric functions.

Alternative Method

Alternatively, one can find the individual values of

\sin\theta

and

\cos\theta

. Since

(\sin\theta - \cos\theta)^2 = \frac{17}{9}

and

\sin\theta\cos\theta = -\frac{4}{9}

, we can use the identity

(\sin\theta + \cos\theta)^2 = 1 + 2\sin\theta\cos\theta = \frac{1}{9}

. In Quadrant II,

\sin\theta > 0

and

\cos\theta < 0

, and since

-\cos\theta > 0

, we have

\sin\theta - \cos\theta = \frac{\sqrt{17}}{3}

. Solving the system of equations for

\sin\theta

and

\cos\theta

yields

\sin\theta = \frac{\sqrt{17} - 1}{6}

(since it must be positive) and

\cos\theta = \frac{-\sqrt{17} - 1}{6}

(since it must be negative). Substituting these into

\tan\theta + \cot\theta = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{\sin\theta}

will yield the same result of

-\frac{9}{4}

, though this method involves significantly more algebraic work.

Estimated Time:3m 0s

Question 20Question →

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

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