Trigonometry Practice Questions — ACT — Page 2

What is the period of the function $f(x) = 4\cos\left(\frac{1}{3}x\right)$ ?

\frac{2\pi}{3}

2\pi

4

6\pi

\frac{4\pi}{3}

Show answer & explanation

Answer: $6\pi$

Answer

The period of the function is

6\pi

.

The standard period of the cosine function

y = \cos(x)

is

2\pi

. For a transformed trigonometric function of the form

y = A\cos(Bx)

, the period is given by the formula

\frac{2\pi}{|B|}

. In the function

f(x) = 4\cos\left(\frac{1}{3}x\right)

, the coefficient of

x

is

B = \frac{1}{3}

. Dividing the standard period

2\pi

by

\frac{1}{3}

yields a period of

2\pi \times 3 = 6\pi

.

Step-by-Step Solution

1

Identify the standard period of the parent cosine function and the coefficient

B

of the variable

x

in the given function.

The parent function is

y = \cos(x)

, which has a standard period of

2\pi

. In the function

f(x) = 4\cos\left(\frac{1}{3}x\right)

, the coefficient of

x

is

B = \frac{1}{3}

.

This sets up the parameters needed for the period formula

\text{Period} = \frac{2\pi}{|B|}

.

2

Substitute the value of

B

into the period formula and simplify.

The period is

\frac{2\pi}{\frac{1}{3}} = 2\pi \times 3 = 6\pi

.

Dividing by a fraction is equivalent to multiplying by its reciprocal, which gives the final horizontal distance for one complete cycle.

Key Concept

Graphs of Trigonometric Functions

Question 22Question →

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 23Question →

In right triangle $PQR$ , the right angle is at vertex $Q$ . If the side lengths are $PQ = 15$ and $QR = 8$ , what is the value of $\tan(P)$ ?

\frac{15}{8}

\frac{15}{17}

\frac{8}{15}

\frac{8}{17}

\frac{17}{8}

Show answer & explanation

Answer: $\frac{8}{15}$

Answer

The tangent of angle

P

is

\frac{8}{15}

.

The tangent of angle

P

is the ratio of the opposite side (

QR = 8

) to the adjacent side (

PQ = 15

). This gives the value

\frac{8}{15}

.

Step-by-Step Solution

1

Identify the reference angle and the sides of the right triangle relative to it.

The reference angle is

P

. The side opposite to angle

P

is

QR

with a length of

8

. The side adjacent to angle

P

is

PQ

with a length of

15

. The hypotenuse is

PR

.

To calculate a trigonometric ratio, we must first determine which sides are opposite, adjacent, and the hypotenuse relative to the target angle.

2

Recall the definition of the tangent ratio in a right triangle.

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side:

\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

.

The question asks for the tangent of angle

P

.

3

Substitute the identified side lengths into the tangent formula.

\tan(P) = \frac{QR}{PQ} = \frac{8}{15}

.

Plugging the lengths of the opposite side (

8

) and the adjacent side (

15

) into the tangent ratio yields the final value.

Key Concept

Right triangle trigonometry ratios (SOHCAHTOA), specifically the tangent ratio definition.

Estimated Time:45s

Question 24Question →

For an angle $\theta$ in the interval $\frac{3\pi}{2} < \theta < 2\pi$ , the expression $\sec\theta - \tan\theta$ is equal to $3$ . What is the value of $\csc\theta + \cot\theta$ ?

Show answer & explanation

Answer: -2

Answer

The value of

\csc\theta + \cot\theta

is

-2

.

By using the difference of squares on the identity

\sec^2\theta - \tan^2\theta = 1

, we get

(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1

. Substituting

\sec\theta - \tan\theta = 3

gives

\sec\theta + \tan\theta = \frac{1}{3}

. Solving the system of equations gives

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

and

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

. Since

\theta

lies in Quadrant IV,

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

and

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

. We then find

\csc\theta = -\frac{5}{4}

and

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

, which sum to

-2

.

Step-by-Step Solution

1

Use the Pythagorean identity

\sec^2\theta - \tan^2\theta = 1

, which factors into

(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1

.

Since

\sec\theta - \tan\theta = 3

, we have

3(\sec\theta + \tan\theta) = 1 \implies \sec\theta + \tan\theta = \frac{1}{3}

.

To establish a second linear equation in terms of

\sec\theta

and

\tan\theta

.

2

Add and subtract the two equations:

\sec\theta - \tan\theta = 3

and

\sec\theta + \tan\theta = \frac{1}{3}

.

Adding them gives

2\sec\theta = \frac{10}{3} \implies \sec\theta = \frac{5}{3}

. Subtracting the first from the second gives

2\tan\theta = -\frac{8}{3} \implies \tan\theta = -\frac{4}{3}

.

To isolate the values of

\sec\theta

and

\tan\theta

.

3

Find

\cos\theta

and

\sin\theta

using

\cos\theta = \frac{1}{\sec\theta}

and

\sin\theta = \tan\theta\cos\theta

.

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

and

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

. Since

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

(Quadrant IV), cosine is positive and sine is negative, which matches these values.

To find the primary trigonometric values needed for the reciprocal functions.

4

Calculate

\csc\theta

and

\cot\theta

using reciprocal identities.

\csc\theta = \frac{1}{\sin\theta} = -\frac{5}{4}

and

\cot\theta = \frac{1}{\tan\theta} = -\frac{3}{4}

.

To obtain the terms of the required sum.

5

Sum the values of

\csc\theta

and

\cot\theta

.

\csc\theta + \cot\theta = -\frac{5}{4} + \left(-\frac{3}{4}\right) = -\frac{8}{4} = -2

.

To obtain the final value requested by the question.

Key Concept

Pythagorean and Reciprocal Trigonometric Identities

Question 25Question →

For an angle $\theta$ such that $\pi < \theta < \frac{3\pi}{2}$ , the equation $\frac{1 - \sin\theta}{\cos\theta} + \frac{\cos\theta}{1 - \sin\theta} = -\frac{10}{3}$ is satisfied. What is the value of $\sin\theta$ ?

-\frac{4}{5}

-\frac{3}{5}

\frac{4}{5}

-\frac{3}{4}

\frac{3}{5}

Show answer & explanation

Answer: $-\frac{4}{5}$

Answer

The value of

\sin\theta

is

-\frac{4}{5}

.

The correct answer is determined by first rewriting the given equation by finding a common denominator, which simplifies the numerator to

2(1-\sin\theta)

using the Pythagorean identity

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

. The term

(1-\sin\theta)

cancels out, resulting in

\frac{2}{\cos\theta} = -\frac{10}{3}

, which implies

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

. In Quadrant III, sine is negative, so using the identity

\sin\theta = -\sqrt{1 - \cos^2\theta}

yields the correct value.

Step-by-Step Solution

1

Find a common denominator to add the fractions on the left-hand side of the equation.

The expression becomes

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

.

To combine the algebraic terms into a single rational expression.

2

Expand the numerator and apply the Pythagorean identity

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

.

The numerator simplifies to

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

.

To reduce the numerator's complexity using fundamental trigonometric identities.

3

Cancel the common factor

(1 - \sin\theta)

from the numerator and denominator, and equate the simplified term to

-\frac{10}{3}

.

The equation simplifies to

\frac{2}{\cos\theta} = -\frac{10}{3}

, which gives

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

.

To solve for the cosine of the angle

\theta

.

4

Use the Pythagorean identity

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

to calculate

\sin\theta

, keeping the quadrant sign rules in mind.

Since

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

(Quadrant III),

\sin\theta

is negative. Therefore,

\sin\theta = -\sqrt{1 - \cos^2\theta} = -\sqrt{1 - \left(-\frac{3}{5}\right)^2} = -\frac{4}{5}

.

To find the correct value and sign of the sine ratio for the given quadrant.

Key Concept

Fundamental Trigonometric Identities

Question 26Question →

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 27Question →

What is the period of the function $f(x) = \tan(3x)$ ?

\frac{2\pi}{3}

3\pi

\frac{\pi}{3}

\pi

3

Show answer & explanation

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

Answer

The period of the function is

\frac{\pi}{3}

.

The parent function

y = \tan(x)

has a standard period of

\pi

. To find the period of a transformed tangent function of the form

y = \tan(Bx)

, the standard period must be divided by the coefficient of

x

, yielding

\frac{\pi}{|B|}

. Substituting

B = 3

gives

\frac{\pi}{3}

.

Step-by-Step Solution

1

Determine the standard period of the parent function.

The parent function is the tangent function,

y = \tan(x)

, which has a standard period of

\pi

.

The tangent function completes one full cycle of its graph between

-\frac{\pi}{2}

and

\frac{\pi}{2}

.

2

Identify the horizontal compression/stretch coefficient from the given equation.

In

f(x) = \tan(3x)

, the coefficient of

x

is

B = 3

.

This coefficient determines how many cycles occur in a standard interval.

3

Calculate the period using the formula for the tangent function.

Period =

\frac{\pi}{|B|} = \frac{\pi}{3}

.

Dividing the standard period of

\pi

by the absolute value of the coefficient

B

gives the compressed period of the transformed function.

Key Concept

The period of a transformed tangent function

y = \tan(Bx)

is given by

\frac{\pi}{|B|}

.

Question 28Question →

A $10$ -foot ladder leans against a vertical wall. The base of the ladder is $6$ feet away from the bottom of the wall, and the top of the ladder reaches a height of $8$ feet up the wall. If $\theta$ is the angle formed between the ladder and the ground, what is the value of $\cos(\theta)$ ?

\frac{3}{4}

\frac{3}{5}

\frac{4}{5}

\frac{4}{3}

\frac{5}{3}

Show answer & explanation

Answer: $\frac{3}{5}$

Answer

The correct answer is

\frac{3}{5}

, which represents the ratio of the adjacent side (

6

feet) to the hypotenuse (

10

feet) for the angle

\theta

formed between the ladder and the ground.

The correct answer is the value

\frac{3}{5}

. The angle

\theta

is formed between the ladder (hypotenuse) and the ground (adjacent side). The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. Here, the adjacent side is

6

feet and the hypotenuse is

10

feet. Therefore,

\cos(\theta) = \frac{6}{10}

, which simplifies to

\frac{3}{5}

.

Step-by-Step Solution

1

Identify the parts of the right triangle relative to the angle

\theta

formed between the ladder and the ground.

The hypotenuse is the length of the ladder (

10

feet). The side adjacent to

\theta

is the distance along the ground from the wall to the base of the ladder (

6

feet). The side opposite to

\theta

is the height up the wall (

8

feet).

To apply trigonometric ratios, we must first map the given dimensions of the scenario to the sides of a right triangle relative to the reference angle.

2

Recall the definition of the cosine ratio in a right triangle.

\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

We need to identify the mathematical definition of cosine to set up the calculation.

3

Substitute the identified side lengths into the cosine ratio and simplify the fraction.

\cos(\theta) = \frac{6}{10} = \frac{3}{5}

Substituting the values of

6

feet for the adjacent side and

10

feet for the hypotenuse gives

\frac{6}{10}

, which simplifies to

\frac{3}{5}

when both the numerator and denominator are divided by their greatest common divisor,

2

.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)

Question 29Question →

An angle $\theta$ lies in the third quadrant, where $\pi < \theta < \frac{3\pi}{2}$ . If $\sin\theta - \cos\theta = -\frac{1}{5}$ , what is the value of the expression $125(\sin^3\theta + \cos^3\theta)$ ?

Show answer & explanation

Answer: -91

Answer

The correct value of the expression is -91.

Squaring the equation

\sin\theta - \cos\theta = -\frac{1}{5}

gives

1 - 2\sin\theta\cos\theta = \frac{1}{25}

, which simplifies to

\sin\theta\cos\theta = \frac{12}{25}

. We then find the square of the sum:

(\sin\theta + \cos\theta)^2 = 1 + 2\sin\theta\cos\theta = 1 + \frac{24}{25} = \frac{49}{25}

. Since the angle is in the third quadrant, both trigonometric functions are negative, so we choose the negative root

\sin\theta + \cos\theta = -\frac{7}{5}

. Factoring the sum of cubes gives

\sin^3\theta + \cos^3\theta = (\sin\theta + \cos\theta)(\sin^2\theta - \sin\theta\cos\theta + \cos^2\theta) = (-\frac{7}{5})(1 - \frac{12}{25}) = -\frac{91}{125}

. Multiplying this by 125 yields -91.

Step-by-Step Solution

1

Square both sides of the equation

\sin\theta - \cos\theta = -\frac{1}{5}

\sin^2\theta - 2\sin\theta\cos\theta + \cos^2\theta = \frac{1}{25} \implies 1 - 2\sin\theta\cos\theta = \frac{1}{25} \implies \sin\theta\cos\theta = \frac{12}{25}

To solve for the product of sine and cosine using the Pythagorean identity

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

.

2

Determine the value of

\sin\theta + \cos\theta

using the identity

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

(\sin\theta + \cos\theta)^2 = 1 + 2(\frac{12}{25}) = \frac{49}{25} \implies \sin\theta + \cos\theta = -\frac{7}{5}

Because

\theta

is in the third quadrant (

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

), both

\sin\theta

and

\cos\theta

must be negative, meaning their sum is also negative.

3

Use the sum of cubes factorization to evaluate

\sin^3\theta + \cos^3\theta

\sin^3\theta + \cos^3\theta = (\sin\theta + \cos\theta)(\sin^2\theta - \sin\theta\cos\theta + \cos^2\theta) = (-\frac{7}{5})(1 - \frac{12}{25}) = -\frac{91}{125}

To express the sum of cubes in terms of the known sum and product of sine and cosine.

4

Multiply the evaluated sum of cubes by 125

125 \times (-\frac{91}{125}) = -91

To compute the final value of the requested expression.

Key Concept

Pythagorean trigonometric identities, quadrant sign analysis, and algebraic factorization of the sum of cubes

Alternative Method

Instead of applying algebraic identities to find the sum of cubes directly, we can solve for the individual values of

\sin\theta

and

\cos\theta

from the system of equations:

\sin\theta - \cos\theta = -\frac{1}{5}

and

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

. Substituting

\sin\theta = \cos\theta - \frac{1}{5}

into the second equation yields

2\cos^2\theta - \frac{2}{5}\cos\theta - \frac{24}{25} = 0

, which factors as

(5\cos\theta + 3)(5\cos\theta - 4) = 0

. Since

\theta

is in Quadrant III,

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

and

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

. Evaluating

125(\sin^3\theta + \cos^3\theta)

directly with these values gives

125((-\frac{4}{5})^3 + (-\frac{3}{5})^3) = 125(-\frac{64}{125} - \frac{27}{125}) = -91

.

Estimated Time:2m 30s

Question 30Question →

The graph of the function $f(x) = a \cos(b(x - c)) + d$ is shown below for constants $a > 0$ , $b > 0$ , $c \in [0, \pi]$ , and $d$ . The graph has a local maximum at $(\frac{\pi}{3}, 5)$ and the nearest local minimum to its right is at $(\frac{5\pi}{6}, -1)$ . What is the y-intercept of the graph of $f(x)$ ?

\frac{1}{2}

\frac{7}{2}

-\frac{5}{2}

2

-\frac{7}{2}

Show answer & explanation

Answer: $\frac{1}{2}$

Answer

The y-intercept of the graph is

\frac{1}{2}

The correct answer is

\frac{1}{2}

. The amplitude of the function is

a = \frac{5 - (-1)}{2} = 3

, and the midline is

d = \frac{5 + (-1)}{2} = 2

. The distance between the consecutive local maximum and local minimum represents half of a period:

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

, meaning the period is

T = \pi

, so

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

. A local maximum occurs when the argument of the cosine function is

0

, so

2(\frac{\pi}{3} - c) = 0 \implies c = \frac{\pi}{3}

. This gives the equation

f(x) = 3 \cos(2(x - \frac{\pi}{3})) + 2

. Finding the y-intercept requires evaluating the function at

x = 0

:

f(0) = 3 \cos(2(0 - \frac{\pi}{3})) + 2 = 3 \cos(-\frac{2\pi}{3}) + 2 = 3(-\frac{1}{2}) + 2 = \frac{1}{2}

.

Step-by-Step Solution

1

Find the amplitude

a

and midline

d

of the trigonometric function.

a = 3

and

d = 2

The amplitude is half the distance between the maximum and minimum values:

a = \frac{5 - (-1)}{2} = 3

. The midline is the average of these values:

d = \frac{5 + (-1)}{2} = 2

.

2

Determine the period

T

and the frequency coefficient

b

.

T = \pi

and

b = 2

The horizontal distance between a consecutive maximum and minimum is half of the period:

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

. Thus, the full period is

T = \pi

. Since

T = \frac{2\pi}{b}

, we find

b = 2

.

3

Determine the horizontal phase shift

c

.

c = \frac{\pi}{3}

A cosine function achieves its maximum when its argument is a multiple of

2\pi

. Since the maximum is at

x = \frac{\pi}{3}

, we set

2(\frac{\pi}{3} - c) = 0

, giving

c = \frac{\pi}{3}

.

4

Evaluate the function at

x = 0

to find the y-intercept.

f(0) = \frac{1}{2}

Substitute the parameters into the function:

f(x) = 3 \cos(2(x - \frac{\pi}{3})) + 2

. Substituting

x = 0

gives

f(0) = 3 \cos(-\frac{2\pi}{3}) + 2 = 3(-\frac{1}{2}) + 2 = \frac{1}{2}

.

Key Concept

Determining the equation of a transformed trigonometric function from key features (maximum and minimum points) and evaluating it.

Estimated Time:3m 0s

Question 31Question →

In right triangle $ABC$ , the right angle is at vertex $B$ , $AB = 12$ , and $BC = 5$ . A line segment $BD$ is drawn perpendicular to the hypotenuse $AC$ such that $D$ lies on $AC$ . From point $D$ , a perpendicular line segment $DE$ is drawn to side $AB$ , where $E$ lies on $AB$ . What is the length of segment $DE$ ?

\frac{300}{169}

\frac{60}{13}

\frac{720}{169}

\frac{144}{13}

\frac{1728}{169}

Show answer & explanation

Answer: $\frac{720}{169}$

Answer

\frac{720}{169}

The correct answer is

\frac{720}{169}

. First, find the hypotenuse of the right triangle

ABC

using the Pythagorean theorem:

AC = \sqrt{AB^2 + BC^2} = \sqrt{12^2 + 5^2} = 13

. In right triangle

ABC

, the sine of angle

A

is

\sin(\angle A) = \frac{BC}{AC} = \frac{5}{13}

. Next, in right triangle

ABD

(which has a right angle at

D

), the length of the altitude

BD

can be found using the sine of angle

A

:

BD = AB \sin(\angle A) = 12 \cdot \frac{5}{13} = \frac{60}{13}

. In right triangle

BDE

(which has a right angle at

E

), the angle

\angle BDE

is equal to

\angle A

because both are complementary to

\angle ABD

(or

\angle EBD

). Therefore, the adjacent side

DE

is found using the cosine of angle

\angle BDE

:

DE = BD \cos(\angle BDE) = BD \cos(\angle A) = \frac{60}{13} \cdot \frac{12}{13} = \frac{720}{169}

.

Step-by-Step Solution

1

Find the hypotenuse

AC

of the right triangle

ABC

using the Pythagorean theorem.

AC = 13

The hypotenuse is needed to find the trigonometric ratios of angle

A

.

2

Determine

\sin(\angle A)

and

\cos(\angle A)

from right triangle

ABC

.

\sin(\angle A) = \frac{5}{13}

and

\cos(\angle A) = \frac{12}{13}

These trigonometric ratios are needed for the calculations in the nested right triangles.

3

Find the length of altitude

BD

in right triangle

ABD

using

\sin(\angle A)

.

BD = \frac{60}{13}

BD

serves as the hypotenuse for the next right triangle

BDE

.

4

Identify that

\angle BDE = \angle A

and calculate the length of

DE

in right triangle

BDE

.

DE = \frac{720}{169}

Since

\angle BDE = \angle A

,

DE = BD \cos(\angle BDE) = BD \cos(\angle A) = \frac{60}{13} \cdot \frac{12}{13} = \frac{720}{169}

.

Key Concept

Applying SOHCAHTOA (sine and cosine definitions) sequentially across nested right triangles by identifying equal angles.

Estimated Time:2m 0s

Question 32Question →

In triangle $ABC$ , the length of side $a$ is $5$ centimeters, the length of side $b$ is $8$ centimeters, and the measure of angle $C$ is $60^\circ$ . What is the length, in centimeters, of side $c$ ?

Show answer & explanation

Answer: 7

Answer

The length of side

c

is

7

centimeters.

Applying the Law of Cosines directly to the given Side-Angle-Side (SAS) triangle yields

c^2 = 5^2 + 8^2 - 2(5)(8) \cos 60^\circ = 25 + 64 - 40 = 49

, which gives

c = 7

.

Step-by-Step Solution

1

Identify the given values and the appropriate formula.

We are given two sides,

a = 5

and

b = 8

, and the included angle

C = 60^\circ

. To find the opposite side

c

, we use the Law of Cosines:

c^2 = a^2 + b^2 - 2ab \cos C

.

The Law of Cosines relates three sides of a triangle to the cosine of one of its angles, which is applicable for Side-Angle-Side (SAS) configurations.

2

Substitute the known values into the Law of Cosines equation.

c^2 = 5^2 + 8^2 - 2(5)(8) \cos 60^\circ

Plugging the given values into the formula allows us to solve for the unknown side

c

.

3

Evaluate the trigonometric and arithmetic terms.

Since

\cos 60^\circ = 0.5

, we get:

c^2 = 25 + 64 - 80(0.5)

c^2 = 89 - 40

c^2 = 49

Simplifying the expression step-by-step leads to the value of

c^2

.

4

Take the square root of both sides to find the side length.

c = \sqrt{49} = 7

Since side lengths must be positive, the square root of

49

gives the exact length of side

c

.

Key Concept

Law of Cosines

Question 33Question →

A coordinate grid contains a right triangle, $XYZ$ , with the right angle located at vertex $Z$ . The horizontal leg $XZ$ has a length of $24$ units, and the vertical leg $YZ$ has a length of $7$ units. What is the value of $\cos(X)$ ?

\frac{7}{25}

\frac{24}{25}

\frac{7}{24}

\frac{24}{7}

\frac{25}{24}

Show answer & explanation

Answer: $\frac{24}{25}$

Answer

The correct answer is the option representing

\frac{24}{25}

.

The cosine of an angle in a right triangle is the ratio of the length of the adjacent leg to the length of the hypotenuse. The hypotenuse of the triangle is calculated as

25

units using the Pythagorean theorem. Relative to angle

X

, the adjacent leg is

XZ = 24

and the hypotenuse is

XY = 25

. Therefore,

\cos(X) = \frac{24}{25}

.

Step-by-Step Solution

1

Use the Pythagorean theorem to calculate the hypotenuse

XY

.

XY = \sqrt{XZ^2 + YZ^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25

.

To calculate any primary trigonometric ratio, we need the lengths of the relevant sides, including the hypotenuse.

2

Identify the adjacent side relative to angle

X

.

The leg adjacent to angle

X

is

XZ = 24

.

Angle

X

is formed by the hypotenuse

XY

and the adjacent leg

XZ

.

3

Apply the definition of cosine (adjacent over hypotenuse).

\cos(X) = \frac{XZ}{XY} = \frac{24}{25}

.

By SOHCAHTOA, cosine is the ratio of the adjacent side length to the hypotenuse length.

Key Concept

Using SOHCAHTOA definitions to find trigonometric ratios in a right triangle.

Estimated Time:1m 0s

Question 34Question →

For an angle $\theta$ such that $\frac{\pi}{2} < \theta < \pi$ , if $\frac{\cos\theta}{1 - \sin\theta} = -3$ , what is the value of $\sin\theta - \tan\theta$ ?

-\frac{8}{15}

\frac{8}{15}

1

\frac{32}{15}

\frac{35}{12}

Show answer & explanation

Answer: $\frac{32}{15}$

Answer

\frac{32}{15}

By multiplying the numerator and denominator of the given expression by

1 + \sin\theta

and utilizing the Pythagorean identity

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

, we find that

\sec\theta + \tan\theta = -3

. Since

\sec^2\theta - \tan^2\theta = 1

, it follows that

\sec\theta - \tan\theta = -\frac{1}{3}

. Solving this system of equations yields

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

(which means

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

) and

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

. In Quadrant II, sine is positive, so

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

. Thus,

\sin\theta - \tan\theta = \frac{4}{5} - \left(-\frac{4}{3}\right) = \frac{12 + 20}{15} = \frac{32}{15}

.

Step-by-Step Solution

1

Multiply the numerator and denominator of the given expression by

1 + \sin\theta

to simplify it.

\frac{\cos\theta(1 + \sin\theta)}{(1 - \sin\theta)(1 + \sin\theta)} = -3 \implies \frac{\cos\theta(1 + \sin\theta)}{1 - \sin^2\theta} = -3

To set up the Pythagorean identity in the denominator.

2

Substitute the Pythagorean identity

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

into the denominator.

\frac{\cos\theta(1 + \sin\theta)}{\cos^2\theta} = \frac{1 + \sin\theta}{\cos\theta} = \sec\theta + \tan\theta = -3

To simplify the expression into basic trigonometric functions.

3

Use the reciprocal identity relationship

\sec^2\theta - \tan^2\theta = 1

to find the difference of secant and tangent.

\sec\theta - \tan\theta = \frac{1}{\sec\theta + \tan\theta} = \frac{1}{-3} = -\frac{1}{3}

To create a system of linear equations for secant and tangent.

4

Solve the system of equations for

\sec\theta

and

\tan\theta

.

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

and

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

Adding the two equations yields

2\sec\theta = -\frac{10}{3} \implies \sec\theta = -\frac{5}{3}

, and subtracting them yields

2\tan\theta = -\frac{8}{3} \implies \tan\theta = -\frac{4}{3}

.

5

Find

\cos\theta

and

\sin\theta

, and calculate the final value of

\sin\theta - \tan\theta

.

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

,

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

, and

\sin\theta - \tan\theta = \frac{4}{5} - \left(-\frac{4}{3}\right) = \frac{32}{15}

Since

\theta

lies in Quadrant II, sine is positive, which is verified by

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

.

Key Concept

Fundamental Trigonometric Identities

Estimated Time:2m 0s

Question 35Question →

A straight skateboard ramp has a vertical height of $9$ feet and a horizontal length of $12$ feet along the ground. The vertical height and horizontal length meet at a right angle. What is the sine of the angle that the ramp makes with the ground?

\frac{3}{4}

\frac{4}{5}

\frac{3}{5}

\frac{4}{3}

\frac{5}{3}

Show answer & explanation

Answer: $\frac{3}{5}$

Answer

The value of the sine of the angle is

\frac{3}{5}

.

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. The opposite side is the vertical height of

9

feet. Using the Pythagorean theorem, the hypotenuse (the length of the ramp) is calculated as

\sqrt{9^2 + 12^2} = 15

feet. Therefore, the sine of the angle is

\frac{9}{15}

, which simplifies to

\frac{3}{5}

.

Step-by-Step Solution

1

Identify the lengths of the legs of the right triangle and calculate the length of the hypotenuse using the Pythagorean theorem.

The hypotenuse (ramp length) is

15

feet.

The vertical height (

9

feet) and horizontal length (

12

feet) form the perpendicular legs of a right triangle. The length of the ramp is the hypotenuse:

\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15

feet.

2

Identify the opposite side and hypotenuse relative to the angle the ramp makes with the ground, then apply the sine ratio definition.

The sine of the angle is

\frac{3}{5}

.

The angle the ramp makes with the ground is at the bottom vertex. The side opposite this angle is the vertical height (

9

feet), and the hypotenuse is the ramp length (

15

feet). The sine ratio is

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15}

, which simplifies to

\frac{3}{5}

.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)

Estimated Time:1m 0s

Question 36Question →

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 37Question →

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 38Question →

In triangle $PQR$ , the measure of angle $P$ is $30^\circ$ , the measure of angle $Q$ is $45^\circ$ , and the length of side $QR$ is $10$ units. Which of the following expressions represents the length, in units, of side $PR$ ?

10\sin(30^\circ)\sin(45^\circ)

\frac{10\sin(30^\circ)}{\sin(45^\circ)}

\frac{10\cos(45^\circ)}{\cos(30^\circ)}

\frac{10\sin(45^\circ)}{\sin(30^\circ)}

\frac{10\cos(30^\circ)}{\sin(45^\circ)}

Show answer & explanation

Answer: $\frac{10\sin(45^\circ)}{\sin(30^\circ)}$

Answer

The length of side

PR

is represented by the expression

\frac{10\sin(45^\circ)}{\sin(30^\circ)}

.

The correct answer is derived by setting up the Law of Sines proportion:

\frac{PR}{\sin(45^\circ)} = \frac{10}{\sin(30^\circ)}

. Multiplying both sides by

\sin(45^\circ)

yields

PR = \frac{10\sin(45^\circ)}{\sin(30^\circ)}

.

Step-by-Step Solution

1

Identify the known values and corresponding angle-side pairs in triangle

PQR

.

Angle

P = 30^\circ

is opposite to side

QR = 10

, and angle

Q = 45^\circ

is opposite to side

PR

.

This allows us to set up the appropriate trigonometric relationship to solve for the unknown side.

2

Apply the Law of Sines to relate the ratios of the side lengths to the sines of their opposite angles.

\frac{PR}{\sin(Q)} = \frac{QR}{\sin(P)}

, which becomes

\frac{PR}{\sin(45^\circ)} = \frac{10}{\sin(30^\circ)}

.

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides.

3

Isolate the variable representing the length of side

PR

.

PR = \frac{10\sin(45^\circ)}{\sin(30^\circ)}

.

Multiply both sides of the equation by

\sin(45^\circ)

to solve for

PR

.

Key Concept

The Law of Sines relates the side lengths of a triangle to the sines of its angles:

\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}

.

Estimated Time:1m 0s

Question 39Question →

For the trigonometric equations on the left, which description on the right correctly matches the graphical features of each equation?

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct matches are: the equation

y = -3\sin\left(2x - \frac{\pi}{3}\right) + 1

matches the description with a period of

\pi

, a range of

[-2, 4]

, and a positive

y

-intercept; the equation

y = 2\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) - 1

matches the description with a period of

4\pi

, a range of

[-3, 1]

, and a phase shift of

\frac{\pi}{2}

units to the left; the equation

y = -\tan\left(3x - \frac{\pi}{2}\right) + 2

matches the description with a period of

\frac{\pi}{3}

that is undefined at the

y

-axis; and the equation

y = 3\sin\left(\frac{2}{3}x + \pi\right) - 2

matches the description with a period of

3\pi

, a range of

[-5, 1]

, and a

y

-intercept of

(0, -2)

.

Each equation is correctly paired with the unique graphical properties determined by calculating its amplitude, period, range, phase shift, and

y

-intercept. The first equation features a period of

\pi

, a range of

[-2, 4]

, and a positive

y

-intercept. The second equation has a period of

4\pi

, a range of

[-3, 1]

, and a phase shift of

\frac{\pi}{2}

units left. The third equation has a period of

\frac{\pi}{3}

and is undefined at the

y

-axis since

x=0

creates a vertical asymptote. The fourth equation has a period of

3\pi

, a range of

[-5, 1]

, and a

y

-intercept of

(0, -2)

.

Step-by-Step Solution

1

Determine the period, range, and y-intercept for the function

y = -3\sin\left(2x - \frac{\pi}{3}\right) + 1

.

The period is

\pi

, range is

[-2, 4]

, and

y

-intercept is

1 + \frac{3\sqrt{3}}{2} > 0

.

The period is

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

, the range is

[1-3, 1+3] = [-2, 4]

, and evaluating at

x=0

yields

y = -3\sin(-\pi/3) + 1 > 0

.

2

Determine the period, range, and phase shift for the function

y = 2\cos\left(\frac{1}{2}x + \frac{\pi}{4}\right) - 1

.

The period is

4\pi

, range is

[-3, 1]

, and phase shift is

\frac{\pi}{2}

units left.

The period is

\frac{2\pi}{1/2} = 4\pi

, the range is

[-1-2, -1+2] = [-3, 1]

, and factoring the argument yields

\frac{1}{2}(x + \frac{\pi}{2})

, giving a shift of

\frac{\pi}{2}

units left.

3

Determine the period and domain restriction for the function

y = -\tan\left(3x - \frac{\pi}{2}\right) + 2

.

The period is

\frac{\pi}{3}

and the function is undefined at

x = 0

(the

y

-axis).

For tangent, the period is

\frac{\pi}{3}

. At

x=0

, the argument is

-\frac{\pi}{2}

, where tangent is undefined, meaning the function has a vertical asymptote at the

y

-axis.

4

Determine the period, range, and y-intercept for the function

y = 3\sin\left(\frac{2}{3}x + \pi\right) - 2

.

The period is

3\pi

, range is

[-5, 1]

, and

y

-intercept is

(0, -2)

.

The period is

\frac{2\pi}{2/3} = 3\pi

, the range is

[-2-3, -2+3] = [-5, 1]

, and at

x=0

,

y = 3\sin(\pi) - 2 = -2

.

Key Concept

Identifying graphs of trigonometric functions from their equations by determining amplitude, period, phase shift, midline, range, and asymptotes.

Question 40Question →

A surveyor stands at point $A$ on flat ground and measures the angle of elevation to the top of a vertical tower, $T$ , at point $C$ to be $30^\circ$ . Another surveyor at point $B$ , which is $100$ meters away from $A$ on the same flat ground, measures $\angle CAB = 40^\circ$ and $\angle CBA = 65^\circ$ . Which of the following expressions represents the height, in meters, of the tower?

\frac{100 \sin(65^\circ) \tan(30^\circ)}{\sin(75^\circ)}

\frac{100 \sin(40^\circ) \tan(30^\circ)}{\sin(75^\circ)}

\frac{100 \sin(65^\circ) \sin(30^\circ)}{\sin(75^\circ)}

\frac{100 \sin(75^\circ) \tan(30^\circ)}{\sin(65^\circ)}

\frac{100 \sin(65^\circ) \tan(30^\circ)}{\sin(105^\circ)}

Show answer & explanation

Answer: $\frac{100 \sin(65^\circ) \tan(30^\circ)}{\sin(75^\circ)}$

Answer

The correct expression is

\frac{100 \sin(65^\circ) \tan(30^\circ)}{\sin(75^\circ)}

The correct expression is derived by first applying the Law of Sines to find the length of the ground segment

AC

, and then using right-triangle trigonometry to determine the height of the tower. In triangle

ABC

, the third angle

\angle ACB

is

180^\circ - (40^\circ + 65^\circ) = 75^\circ

. The Law of Sines gives

\frac{AC}{\sin(65^\circ)} = \frac{100}{\sin(75^\circ)}

, which simplifies to

AC = \frac{100\sin(65^\circ)}{\sin(75^\circ)}

. Since the tower is vertical, triangle

ACT

is a right triangle with

\tan(30^\circ) = \frac{h}{AC}

. Substituting

AC

yields

h = \frac{100\sin(65^\circ)\tan(30^\circ)}{\sin(75^\circ)}

.

Step-by-Step Solution

1

Calculate the measure of the third angle in the ground triangle

\triangle ABC

.

\angle ACB = 180^\circ - (40^\circ + 65^\circ) = 75^\circ

The sum of angles in any triangle must equal

180^\circ

.

2

Apply the Law of Sines to find the distance from point

A

to the base of the tower at point

C

(

AC

).

\frac{AC}{\sin(65^\circ)} = \frac{100}{\sin(75^\circ)} \Rightarrow AC = \frac{100 \sin(65^\circ)}{\sin(75^\circ)}

The Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant in a triangle.

3

Use the right-triangle trigonometric ratio for the vertical tower height

h

from point

A

.

\tan(30^\circ) = \frac{h}{AC} \Rightarrow h = AC \tan(30^\circ)

In right triangle

\triangle ACT

, the tangent of the angle of elevation is the ratio of the opposite side (height

h

) to the adjacent side (

AC

).

4

Substitute the expression for

AC

into the equation for

h

.

h = \frac{100 \sin(65^\circ) \tan(30^\circ)}{\sin(75^\circ)}

Replacing

AC

with its equivalent algebraic expression yields the final height in terms of the given parameters.

Key Concept

Applying the Law of Sines to find a missing side length in a non-right triangle and then using right-triangle trigonometric ratios to solve a 3D geometry problem.

Estimated Time:2m 0s

Trigonometry

Answer

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Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes