Trigonometric Ratios and Identities Practice Questions — SAT

Question 1Question →

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\tan(P) = \frac{4}{3}$ , what is the value of $\sin(P)$ ?

\frac{4}{5}

\frac{3}{5}

\frac{3}{4}

\frac{5}{4}

Show answer & explanation

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

Answer

\frac{4}{5}

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Given that

\tan(P) = \frac{4}{3}

, we can set the length of the side opposite to angle P as

4

and the adjacent side as

3

. Using the Pythagorean theorem, the hypotenuse is

\sqrt{3^2 + 4^2} = 5

. Since the sine of an angle is the ratio of the opposite side to the hypotenuse,

\sin(P) = \frac{4}{5}

.

Step-by-Step Solution

1

Identify the relationship between the tangent ratio and the sides of the right triangle.

The tangent of angle P is defined as the opposite side divided by the adjacent side. Since

\tan(P) = \frac{4}{3}

, we can set the opposite side to

4

and the adjacent side to

3

.

This allows us to establish the relative lengths of the sides of the right triangle.

2

Calculate the length of the hypotenuse using the Pythagorean theorem.

The hypotenuse is

\sqrt{3^2 + 4^2} = \sqrt{25} = 5

.

The hypotenuse is required to calculate the sine ratio.

3

Determine the sine of angle P.

The sine of angle P is defined as the opposite side divided by the hypotenuse, which is

\frac{4}{5}

.

This yields the final requested trigonometric ratio.

Key Concept

Using basic trigonometric ratios (SOH CAH TOA) and the Pythagorean theorem to evaluate trigonometric ratios in a right triangle.

Estimated Time:45s

Question 2Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\sin(A) - \cos(A) = \frac{1}{5}$ , what is the value of $\sin(A)\sin(B)$ ?

\frac{13}{25}

\frac{16}{25}

\frac{12}{25}

\frac{24}{25}

Show answer & explanation

Answer: $\frac{12}{25}$

Answer

\frac{12}{25}

By the co-function identity for complementary angles in a right triangle,

\sin(B) = \cos(A)

, which means the expression

\sin(A)\sin(B)

is equivalent to

\sin(A)\cos(A)

. Squaring the equation

\sin(A) - \cos(A) = \frac{1}{5}

gives

\sin^2(A) - 2\sin(A)\cos(A) + \cos^2(A) = \frac{1}{25}

. Substituting the Pythagorean identity

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

simplifies this to

1 - 2\sin(A)\cos(A) = \frac{1}{25}

. Solving for

\sin(A)\cos(A)

yields

2\sin(A)\cos(A) = \frac{24}{25}

, or

\sin(A)\cos(A) = \frac{12}{25}

.

Step-by-Step Solution

1

Express the target expression in terms of a single angle using complementary relationships.

Since angle

C = 90^\circ

, angles

A

and

B

are complementary, so

\sin(B) = \cos(A)

. Thus,

\sin(A)\sin(B) = \sin(A)\cos(A)

.

In any right triangle, the sine of one acute angle equals the cosine of the other acute angle.

2

Square both sides of the given equation

\sin(A) - \cos(A) = \frac{1}{5}

.

(\sin(A) - \cos(A))^2 = \left(\frac{1}{5}\right)^2 \implies \sin^2(A) - 2\sin(A)\cos(A) + \cos^2(A) = \frac{1}{25}

.

Squaring the difference allows us to introduce the product term

\sin(A)\cos(A)

and the squared terms.

3

Apply the Pythagorean identity to simplify the equation.

Since

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

, the equation becomes

1 - 2\sin(A)\cos(A) = \frac{1}{25}

.

The sum of the squares of sine and cosine of the same angle is always equal to 1.

4

Solve for the product

\sin(A)\cos(A)

.

2\sin(A)\cos(A) = 1 - \frac{1}{25} = \frac{24}{25} \implies \sin(A)\cos(A) = \frac{12}{25}

.

Subtracting

1/25

from

1

and dividing the resulting fraction by

2

isolates the target product.

Key Concept

Pythagorean identity and co-function relationships in right triangles

Question 3Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . The acute angles $A$ and $C$ satisfy $\sin(A) = \frac{k}{3}$ and $\cos(C) = \frac{4}{k+4}$ for some positive constant $k$ . What is the value of $k$ ?

6

8

2

12

Show answer & explanation

Answer: 2

Answer

2

In right triangle

ABC

, the measure of angle

B

is

90^\circ

, which means the acute angles

A

and

C

are complementary. Therefore, the co-function identity states that

\sin(A) = \cos(C)

. Equating the given expressions yields

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

. Cross-multiplying results in

k(k+4) = 12

, which simplifies to

k^2 + 4k - 12 = 0

. Factoring this equation gives

(k+6)(k-2) = 0

. This quadratic has solutions

k = -6

and

k = 2

. Since

k

must be positive, the value of

k

is

2

.

Step-by-Step Solution

1

Identify the relationship between the acute angles in a right triangle.

Since the measure of angle

B

is

90^\circ

, the sum of angles

A

and

C

is

90^\circ

, which means they are complementary angles. By the co-function identity,

\sin(A) = \cos(C)

.

In a right triangle, the sine of one acute angle equals the cosine of the other acute angle.

2

Equate the expressions for

\sin(A)

and

\cos(C)

and set up the equation for

k

.

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

This sets up the algebraic relation to solve for

k

using the given trigonometric expressions.

3

Solve the algebraic equation for

k

.

Cross-multiplying gives

k(k+4) = 12

, which expands to

k^2 + 4k = 12

. Subtracting

12

from both sides results in the quadratic equation

k^2 + 4k - 12 = 0

. Factoring this equation yields

(k+6)(k-2) = 0

.

Cross-multiplication eliminates the denominators, converting the equation to a quadratic form that can be solved by factoring.

4

Determine the valid positive value for

k

.

The solutions to the equation

(k+6)(k-2) = 0

are

k = -6

and

k = 2

. Since

k

must be a positive constant, we select

k = 2

.

The problem specifies that

k

is a positive constant, so the negative solution is discarded.

Key Concept

Co-function identity relating sine and cosine of complementary angles

Estimated Time:1m 30s

Question 4Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\cos(A) = k \sin(A)$ for some positive constant $k$ , which of the following expressions represents $\cos(B)$ in terms of $k$ ?

\frac{k}{\sqrt{1 + k^2}}

\frac{1}{\sqrt{1 - k^2}}

\frac{1}{\sqrt{1 + k^2}}

\sqrt{1 + k^2}

Show answer & explanation

Answer: $\frac{1}{\sqrt{1 + k^2}}$

Answer

The expression that represents

\cos(B)

is

\frac{1}{\sqrt{1 + k^2}}

.

In right triangle

ABC

with right angle

C

, the acute angles

A

and

B

are complementary (

A + B = 90^\circ

). By the co-function identity,

\cos(B) = \sin(A)

. Using the given relation

\cos(A) = k \sin(A)

and substituting it into the Pythagorean identity

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

, we get

\sin^2(A) + (k \sin(A))^2 = 1 \implies \sin^2(A)(1 + k^2) = 1

. Solving for

\sin(A)

gives

\sin(A) = \frac{1}{\sqrt{1 + k^2}}

because

\sin(A)

must be positive for an acute angle. Since

\cos(B) = \sin(A)

, we conclude that the correct expression is

\frac{1}{\sqrt{1 + k^2}}

.

Step-by-Step Solution

1

Use the relationship between the acute angles in right triangle

ABC

.

Since angle

C

is

90^\circ

, angles

A

and

B

are complementary. Thus,

\cos(B) = \sin(A)

.

This allows us to convert the target term

\cos(B)

into

\sin(A)

, which can be related to the given equation.

2

Substitute the given relation

\cos(A) = k \sin(A)

into the Pythagorean identity.

Using

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

, we substitute to get

\sin^2(A) + (k \sin(A))^2 = 1

, which simplifies to

\sin^2(A)(1 + k^2) = 1

.

This sets up a single equation containing only

\sin(A)

and the constant

k

.

3

Solve for

\sin(A)

and substitute back to find

\cos(B)

.

Solving for

\sin(A)

gives

\sin(A) = \frac{1}{\sqrt{1 + k^2}}

(since

\sin(A) > 0

for acute angle

A

). Since

\cos(B) = \sin(A)

, we have

\cos(B) = \frac{1}{\sqrt{1 + k^2}}

.

This yields the final value of

\cos(B)

in terms of

k

.

Key Concept

Applying complementary angle trigonometric identities (co-functions) and the Pythagorean identity in a right triangle.

Alternative Method

Alternatively, you can model this by setting up a right triangle. Since

\cos(A) = k \sin(A)

, we can divide both sides by

\sin(A)

to get

\cot(A) = k

, which means

\tan(A) = \frac{1}{k}

. In a right triangle

ABC

with right angle

C

,

\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AC} = \frac{1}{k}

. Let

BC = 1

and

AC = k

. By the Pythagorean theorem, the hypotenuse

AB = \sqrt{1^2 + k^2} = \sqrt{1 + k^2}

. Then,

\cos(B) = \frac{\text{adjacent to } B}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{1}{\sqrt{1 + k^2}}

.

Estimated Time:2m 0s

Question 5Question →

If $\cos(\theta) = \frac{7}{25}$ and $0 < \theta < \frac{\pi}{2}$ , what is the value of $\sin\left(\frac{\pi}{2} - \theta\right)$ ?

\frac{24}{25}

\frac{7}{25}

\frac{25}{7}

\frac{18}{25}

Show answer & explanation

Answer: $\frac{7}{25}$

Answer

\frac{7}{25}

The correct answer is

\frac{7}{25}

because the co-function identity states that for any angle

\theta

, the sine of its complement,

\frac{\pi}{2} - \theta

, is equal to the cosine of

\theta

. Given that

\cos(\theta) = \frac{7}{25}

, it follows that

\sin\left(\frac{\pi}{2} - \theta\right) = \frac{7}{25}

.

Step-by-Step Solution

1

Identify the relevant trigonometric identity.

Use the co-function identity

\sin\left(\frac{\pi}{2} - x\right) = \cos(x)

.

Since the angle is given in radians, the complementary angle identity relates the sine of the complement to the cosine of the original angle.

2

Substitute the given value.

Since

\cos(\theta) = \frac{7}{25}

, then

\sin\left(\frac{\pi}{2} - \theta\right) = \frac{7}{25}

.

Direct substitution of the given cosine value into the identity yields the final answer.

Key Concept

Co-function identities relate the sine and cosine of complementary angles.

Question 6Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . If $\sin(A) = \frac{5}{13}$ , what is the value of $\cos(C)$ ?

\frac{5}{13}

\frac{12}{13}

\frac{13}{5}

\frac{8}{13}

Show answer & explanation

Answer: $\frac{5}{13}$

Answer

\frac{5}{13}

The correct answer is

\frac{5}{13}

. In any right triangle

ABC

where the right angle is at

B

, the two acute angles

A

and

C

are complementary, meaning

A + C = 90^\circ

. The co-function identity states that the sine of an acute angle is equal to the cosine of its complement, or

\sin(A) = \cos(90^\circ - A) = \cos(C)

. Given that

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

, the value of

\cos(C)

must also be

\frac{5}{13}

.

Step-by-Step Solution

1

Identify the relationship between the acute angles in a right triangle.

Since the measure of angle

B

is

90^\circ

, the sum of the measures of angles

A

and

C

must be

90^\circ

. Thus, angles

A

and

C

are complementary.

The sum of angles in any triangle is

180^\circ

.

2

Apply the co-function identity for complementary angles.

For any two complementary angles

A

and

C

, the identity

\cos(C) = \sin(A)

holds true.

The sine of an angle is the ratio of the opposite side to the hypotenuse, which is the same as the ratio of the adjacent side of its complement to the hypotenuse (cosine of the complement).

3

Substitute the given value to find

\cos(C)

.

Since

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

, it follows that

\cos(C) = \frac{5}{13}

.

Direct substitution into the identity

\cos(C) = \sin(A)

.

Key Concept

Co-function identities for complementary angles in a right triangle

Question 7Question →

If $\sin(x^\circ) = \cos(38^\circ)$ , where $0 < x < 90$ , what is the value of $x$ ?

Show answer & explanation

Answer: 52

Answer

The value of

x

is 52.

Applying the co-function identity

\sin(\theta) = \cos(90^\circ - \theta)

allows us to set the argument of the sine function as the complement of the cosine's argument. Thus,

x = 90 - 38 = 52

.

Step-by-Step Solution

1

Identify the relevant trigonometric identity.

The co-function identity

\sin(x^\circ) = \cos(90^\circ - x^\circ)

is appropriate here.

We need to relate the sine of one angle to the cosine of another angle.

2

Equate the angle arguments using the identity.

\sin(x^\circ) = \cos(90^\circ - x^\circ) = \cos(38^\circ)

, which implies

90 - x = 38

.

For acute angles, if their cosine values are equal, the angles themselves must be equal.

3

Solve the linear equation for

x

.

x = 90 - 38 = 52

.

Isolating

x

gives the final answer.

Key Concept

Co-function identities relate the sine of an angle to the cosine of its complement:

\sin(\theta) = \cos(90^\circ - \theta)

.

Question 8Question →

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ , $XY = 12$ , and $YZ = 5$ . What is the value of $\cos(X) - \sin(X)$ ?

\frac{7}{13}

\frac{17}{13}

-\frac{7}{13}

\frac{7}{12}

Show answer & explanation

Answer: $\frac{7}{13}$

Answer

\frac{7}{13}

The length of the hypotenuse

XZ

is first found using the Pythagorean theorem:

XZ = \sqrt{12^2 + 5^2} = 13

. Using the definitions of the trigonometric ratios,

\cos(X) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}

and

\sin(X) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}

. The difference is

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

.

Step-by-Step Solution

1

Find the length of the hypotenuse

XZ

using the Pythagorean theorem.

XZ = \sqrt{XY^2 + YZ^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

.

The hypotenuse is needed as the denominator for the sine and cosine ratios.

2

Determine the values of

\cos(X)

and

\sin(X)

.

\cos(X) = \frac{XY}{XZ} = \frac{12}{13}

and

\sin(X) = \frac{YZ}{XZ} = \frac{5}{13}

.

By definition,

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

and

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

.

3

Calculate the difference

\cos(X) - \sin(X)

.

\cos(X) - \sin(X) = \frac{12}{13} - \frac{5}{13} = \frac{7}{13}

.

This is the final subtraction requested by the question.

Key Concept

Calculating trigonometric ratios in a right triangle and applying basic operations.

Question 9Question →

If $\sin(\theta) = \frac{\sqrt{5}}{3}$ and $\theta$ is an acute angle, what is the value of $\tan^2(\theta)$ ?

Show answer & explanation

Answer: 1.25

Answer

The correct answer is 1.25 (or the equivalent fraction 5/4).

Using the Pythagorean identity

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

, we find

\cos^2(\theta) = 1 - \frac{5}{9} = \frac{4}{9}

. Since

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

, it follows that

\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)} = \frac{5/9}{4/9} = 1.25

.

Step-by-Step Solution

1

Calculate the square of the cosine of the angle using the Pythagorean identity.

\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(\frac{\sqrt{5}}{3}\right)^2 = 1 - \frac{5}{9} = \frac{4}{9}

The Pythagorean identity states that

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

for any angle

\theta

.

2

Express the tangent squared in terms of sine squared and cosine squared.

\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}

By definition, the tangent function is the ratio of sine to cosine, so

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

.

3

Substitute the values of

\sin^2(\theta)

and

\cos^2(\theta)

to solve for

\tan^2(\theta)

.

\tan^2(\theta) = \frac{5/9}{4/9} = \frac{5}{4} = 1.25

Plugging in the squared ratios and simplifying gives the final evaluation.

Key Concept

Pythagorean trigonometric identity and quotient identity of tangent

Question 10Question →

In right triangle $JKL$ , the measure of angle $K$ is $90^\circ$ . If $\cos(J) = 0.28$ , what is the value of $\sin(L)$ ?

Show answer & explanation

Answer: 0.28

Answer

The correct answer is 0.28 (or the equivalent fraction 7/25).

In a right triangle, the two acute angles are complementary, meaning they add up to

90^\circ

. The co-function identity states that the sine of an acute angle is equal to the cosine of its complement. Therefore, in right triangle

JKL

with right angle

K

,

\sin(L) = \cos(J)

. Given that

\cos(J) = 0.28

, the value of

\sin(L)

must also be

0.28

.

Step-by-Step Solution

1

Determine the relationship between angles

J

and

L

.

J + L = 90^\circ

Since the sum of angles in a triangle is

180^\circ

and angle

K

is

90^\circ

, the sum of the remaining two angles must be

180^\circ - 90^\circ = 90^\circ

.

2

Use the co-function trigonometric identity.

\sin(L) = \cos(J)

The sine of an acute angle in a right triangle is equal to the cosine of its complementary angle.

3

Substitute the given value of

\cos(J)

into the identity.

\sin(L) = 0.28

We are given that

\cos(J) = 0.28

.

Key Concept

Complementary angle trigonometric identity (co-function identity)

Question 11Question →

An acute angle $\theta$ in a right triangle satisfies $\cos(\theta) = \frac{3}{5}$ . If $\phi$ is the other acute angle of the triangle, what is the value of $\sin(\phi) \tan(\theta)$ ?

\frac{16}{15}

\frac{9}{20}

\frac{4}{5}

\frac{12}{25}

Show answer & explanation

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

Answer

four-fifths

The correct answer is four-fifths. Because

\theta

and

\phi

are complementary angles in a right triangle,

\sin(\phi)

equals

\cos(\theta)

, which is given as three-fifths. Using the Pythagorean identity,

\sin(\theta)

is four-fifths, making

\tan(\theta)

equal to four-thirds. Multiplying

\sin(\phi)

and

\tan(\theta)

gives three-fifths times four-thirds, which simplifies to four-fifths.

Step-by-Step Solution

1

Use the complementary angle relationship to find

\sin(\phi)

.

\sin(\phi) = \cos(\theta) = \frac{3}{5}

The two acute angles in a right triangle,

\theta

and

\phi

, sum to

90^\circ

. Therefore, the sine of one angle equals the cosine of the other.

2

Find

\sin(\theta)

using the Pythagorean identity.

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

Since

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

, we have

\sin^2(\theta) + \left(\frac{3}{5}\right)^2 = 1

, which gives

\sin^2(\theta) = \frac{16}{25}

. Because

\theta

is acute,

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

.

3

Calculate

\tan(\theta)

using the quotient identity.

\tan(\theta) = \frac{4}{3}

By definition,

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{4/5}{3/5} = \frac{4}{3}

.

4

Multiply

\sin(\phi)

by

\tan(\theta)

.

\sin(\phi) \tan(\theta) = \frac{4}{5}

Substitute the found values:

\sin(\phi) \tan(\theta) = \frac{3}{5} \times \frac{4}{3} = \frac{4}{5}

.

Key Concept

Trigonometric co-function identities and quotient identities in right triangles

Question 12Question →

An acute angle $\theta$ satisfies the equation $\sin(\theta) = 0.6$ . What is the value of $\cos(\theta)$ ?

Show answer & explanation

Answer: 0.8

Answer

The value of

\cos(\theta)

is 0.8.

Using the fundamental Pythagorean trigonometric identity

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

, we substitute the given value

\sin(\theta) = 0.6

to obtain

(0.6)^2 + \cos^2(\theta) = 1

, which simplifies to

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

. Subtracting

0.36

from both sides yields

\cos^2(\theta) = 0.64

. Taking the positive square root because

\theta

is an acute angle gives

\cos(\theta) = 0.8

.

Step-by-Step Solution

1

State the Pythagorean trigonometric identity

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

This identity relates the sine and cosine of any angle.

2

Substitute the value of

\sin(\theta)

into the identity

(0.6)^2 + \cos^2(\theta) = 1

We are given that

\sin(\theta) = 0.6

.

3

Solve for

\cos^2(\theta)

\cos^2(\theta) = 0.64

Subtracting

0.36

from

1

isolates the squared cosine term.

4

Take the square root of both sides

\cos(\theta) = 0.8

Since

\theta

is an acute angle, the value of

\cos(\theta)

must be positive.

Key Concept

Pythagorean Identity

Question 13Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\cos(A) = 3\sin(A)$ , what is the value of $\tan(B)$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is

3

.

Because angle

C

is

90^\circ

in right triangle

ABC

, the acute angles

A

and

B

are complementary (

A + B = 90^\circ

). By the co-function identities,

\sin(B) = \cos(A)

and

\cos(B) = \sin(A)

. The tangent of

B

is defined as

\tan(B) = \frac{\sin(B)}{\cos(B)}

. Substituting the co-function identities gives

\tan(B) = \frac{\cos(A)}{\sin(A)}

. Since we are given that

\cos(A) = 3\sin(A)

, we substitute this expression into the numerator to get

\tan(B) = \frac{3\sin(A)}{\sin(A)} = 3

.

Step-by-Step Solution

1

Determine the relationship between the acute angles in right triangle

ABC

.

\sin(B) = \cos(A)

and

\cos(B) = \sin(A)

Since angle

C

is

90^\circ

, the other two angles

A

and

B

must sum to

90^\circ

(they are complementary angles).

2

Express

\tan(B)

in terms of the trigonometric ratios of angle

A

.

\tan(B) = \frac{\cos(A)}{\sin(A)}

By definition, the tangent of angle

B

is the ratio of its sine to its cosine, which yields

\frac{\cos(A)}{\sin(A)}

after substituting the complementary angle relations.

3

Substitute the given relation

\cos(A) = 3\sin(A)

into the expression for

\tan(B)

.

\tan(B) = \frac{3\sin(A)}{\sin(A)}

Substituting the value of

\cos(A)

allows us to simplify the fraction by expressing both terms with

\sin(A)

.

4

Simplify the fraction to get the final numerical value.

\tan(B) = 3

The term

\sin(A)

cancels out from the numerator and denominator since

A

is an acute angle and

\sin(A) \neq 0

.

Key Concept

Co-function identities and trigonometric definitions in a right triangle.

Alternative Method

Alternatively, construct a right triangle where the side opposite to angle

A

has length

1

. Since

\cos(A) = 3\sin(A)

, the ratio of the adjacent side to the hypotenuse is

3

times the ratio of the opposite side to the hypotenuse, meaning the side adjacent to angle

A

must have length

3

. Because angle

B

is the complement of angle

A

, the side opposite to angle

B

is the side adjacent to angle

A

(which is

3

), and the side adjacent to angle

B

is the side opposite to angle

A

(which is

1

). Therefore,

\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{1} = 3

.

Estimated Time:1m 30s

Question 14Question →

In right triangle $RST$ , the measure of angle $T$ is $90^\circ$ and the length of the hypotenuse $RS$ is $10$ . If $\tan(R) + \tan(S) = \frac{5}{2}$ , what is the area of triangle $RST$ ?

10

20

40

50

Show answer & explanation

Answer: 20

Answer

20

Let the lengths of the legs of right triangle

RST

be

RT = x

and

ST = y

. The tangent ratios are

\tan(R) = \frac{y}{x}

and

\tan(S) = \frac{x}{y}

. Substituting these into the given equation gives

\frac{y}{x} + \frac{x}{y} = \frac{5}{2}

, which simplifies to

\frac{x^2 + y^2}{xy} = \frac{5}{2}

. By the Pythagorean theorem,

x^2 + y^2 = 10^2 = 100

. Thus,

\frac{100}{xy} = \frac{5}{2}

, which solves to

xy = 40

. The area of the triangle is

\frac{1}{2}xy = \frac{1}{2}(40) = 20

.

Step-by-Step Solution

1

Express the tangent of angles

R

and

S

in terms of the leg lengths

RT

and

ST

.

\tan(R) = \frac{ST}{RT}

and

\tan(S) = \frac{RT}{ST}

.

By definition, the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.

2

Substitute these expressions into the given equation and simplify using the Pythagorean theorem.

\frac{ST}{RT} + \frac{RT}{ST} = \frac{ST^2 + RT^2}{RT \cdot ST} = \frac{100}{RT \cdot ST} = \frac{5}{2}

.

Finding a common denominator yields the sum of the squares of the legs in the numerator, which equals the square of the hypotenuse (

RS^2 = 10^2 = 100

).

3

Solve for the product of the legs

RT \cdot ST

and calculate the area of the triangle.

RT \cdot ST = 40

, so

\text{Area} = \frac{1}{2}(RT \cdot ST) = \frac{1}{2}(40) = 20

.

The area of a right triangle is half the product of its perpendicular legs.

Key Concept

Using trigonometric ratios, complementary angles, and the Pythagorean theorem to calculate the area of a right triangle.

Question 15Question →

In a right triangle, one of the acute angles is $\theta$ . If $\tan(\theta) = \frac{x}{y}$ , where $x$ and $y$ are positive constants, what is the value of $\cos\left(\frac{\pi}{2} - \theta\right)$ in terms of $x$ and $y$ ?

\frac{y}{\sqrt{x^2 + y^2}}

\frac{x}{\sqrt{x^2 + y^2}}

\frac{y}{x}

\frac{x}{x^2 + y^2}

Show answer & explanation

Answer: $\frac{x}{\sqrt{x^2 + y^2}}$

Answer

\frac{x}{\sqrt{x^2 + y^2}}

Using the cofunction identity,

\cos\left(\frac{\pi}{2} - \theta\right)

is equivalent to

\sin(\theta)

. For an angle

\theta

in a right triangle,

\tan(\theta) = \frac{x}{y}

defines the ratio of the opposite side (

x

) to the adjacent side (

y

). Applying the Pythagorean theorem, the hypotenuse is

\sqrt{x^2 + y^2}

. Therefore,

\sin(\theta)

, which is the ratio of the opposite side to the hypotenuse, is equal to

\frac{x}{\sqrt{x^2 + y^2}}

.

Step-by-Step Solution

1

Apply the cofunction identity to rewrite the target expression.

\cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta)

By the cofunction identities, the sine of an acute angle is equal to the cosine of its complementary angle.

2

Relate the given tangent ratio to the side lengths of a right triangle.

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{y}

, so the opposite side can be represented as

x

and the adjacent side as

y

.

The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.

3

Apply the Pythagorean theorem to calculate the hypotenuse of the triangle.

\text{Hypotenuse} = \sqrt{x^2 + y^2}

The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

4

Express the sine of the angle as a ratio of the opposite side to the hypotenuse.

\sin(\theta) = \frac{x}{\sqrt{x^2 + y^2}}

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

Key Concept

Cofunction identities and right triangle trigonometric ratios

Estimated Time:1m 30s

Question 16Question →

For an acute angle $\theta$ , $\cos(\theta) = \frac{5}{13}$ . What is the value of $5\tan(\theta) + 13\sin(\theta)$ ?

Show answer & explanation

Answer: 24

Answer

24

Using the Pythagorean identity

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

with

\cos(\theta) = \frac{5}{13}

gives

\sin(\theta) = \frac{12}{13}

because

\theta

is an acute angle. The quotient identity gives

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{12}{5}

. Substituting these ratios into the expression

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

.

Step-by-Step Solution

1

Find the value of

\sin(\theta)

using the Pythagorean identity.

\sin(\theta) = \frac{12}{13}

Since

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

and

\theta

is an acute angle, the sine value is positive:

\sin(\theta) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}

.

2

Find the value of

\tan(\theta)

using the quotient identity.

\tan(\theta) = \frac{12}{5}

By definition,

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

. Substituting the known values yields

\tan(\theta) = \frac{12/13}{5/13} = \frac{12}{5}

.

3

Substitute the trigonometric ratios into the given expression and simplify.

24

Substituting the values of

\tan(\theta)

and

\sin(\theta)

into

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

.

Key Concept

Trigonometric ratios and identities, specifically the Pythagorean identity

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

and the definition of tangent as

\frac{\sin(\theta)}{\cos(\theta)}

.

Alternative Method

Alternatively, draw a right triangle with an acute angle

\theta

. Since

\cos(\theta) = \frac{5}{13} = \frac{\text{adjacent}}{\text{hypotenuse}}

, label the adjacent side as 5 and the hypotenuse as 13. By the Pythagorean theorem, the opposite side is

\sqrt{13^2 - 5^2} = 12

. From this triangle,

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}

and

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}

. Substituting these values into the expression gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 24

.

Estimated Time:1m 30s

Question 17Question →

For an acute angle $\theta$ , the sum of its sine and cosine is $\frac{\sqrt{7}}{2}$ . What is the value of $\sin^3(\theta) + \cos^3(\theta)$ ?

\frac{5\sqrt{7}}{4}

\frac{11\sqrt{7}}{16}

\frac{5\sqrt{7}}{16}

\frac{7\sqrt{7}}{8}

Show answer & explanation

Answer: $\frac{5\sqrt{7}}{16}$

Answer

The correct answer is

\frac{5\sqrt{7}}{16}

.

The correct answer is

\frac{5\sqrt{7}}{16}

. Squaring the given equation

\sin(\theta) + \cos(\theta) = \frac{\sqrt{7}}{2}

gives

\sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{7}{4}

. Substituting the Pythagorean identity

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

gives

1 + 2\sin(\theta)\cos(\theta) = \frac{7}{4}

, which simplifies to

\sin(\theta)\cos(\theta) = \frac{3}{8}

. Factoring the sum of cubes yields

\sin^3(\theta) + \cos^3(\theta) = (\sin(\theta) + \cos(\theta))(1 - \sin(\theta)\cos(\theta))

. Substituting the values results in

\left(\frac{\sqrt{7}}{2}\right)\left(1 - \frac{3}{8}\right) = \frac{5\sqrt{7}}{16}

.

Step-by-Step Solution

1

Square both sides of the given equation

\sin(\theta) + \cos(\theta) = \frac{\sqrt{7}}{2}

.

(\sin(\theta) + \cos(\theta))^2 = \sin^2(\theta) + 2\sin(\theta)\cos(\theta) + \cos^2(\theta) = \frac{7}{4}

.

This sets up the expression to apply the Pythagorean trigonometric identity.

2

Substitute the Pythagorean identity

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

and solve for the product term.

1 + 2\sin(\theta)\cos(\theta) = \frac{7}{4} \implies 2\sin(\theta)\cos(\theta) = \frac{3}{4} \implies \sin(\theta)\cos(\theta) = \frac{3}{8}

.

This determines the value of the product of the sine and cosine ratios.

3

Apply the sum of cubes factorization formula to

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

.

This expresses the target quantity in terms of known sum and product values.

4

Substitute the values of the sum and product into the factored expression and simplify.

\sin^3(\theta) + \cos^3(\theta) = \left(\frac{\sqrt{7}}{2}\right)\left(1 - \frac{3}{8}\right) = \left(\frac{\sqrt{7}}{2}\right)\left(\frac{5}{8}\right) = \frac{5\sqrt{7}}{16}

.

This yields the final simplified numerical ratio.

Key Concept

Pythagorean Identity and Algebraic Manipulation of Trigonometric Ratios

Question 18Question →

For an acute angle $\theta$ , the equation $2\cos^2(\theta) - 5\sin(\theta) + 1 = 0$ is true. What is the value of $\sin(\theta)$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

\sin(\theta)

is

0.5

(or

\frac{1}{2}

)

By applying the Pythagorean identity

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

, the equation

2\cos^2(\theta) - 5\sin(\theta) + 1 = 0

can be rewritten entirely in terms of

\sin(\theta)

, yielding

-2\sin^2(\theta) - 5\sin(\theta) + 3 = 0

. Multiplying by

-1

gives the standard quadratic equation

2\sin^2(\theta) + 5\sin(\theta) - 3 = 0

, which factors as

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

. Solving for

\sin(\theta)

yields

\sin(\theta) = 0.5

or

\sin(\theta) = -3

. Since the sine value of any angle must be in the range

[-1, 1]

and the sine of an acute angle must be positive,

\sin(\theta) = 0.5

is the only valid solution.

Step-by-Step Solution

1

Apply the Pythagorean identity to rewrite the cosine term.

2(1 - \sin^2(\theta)) - 5\sin(\theta) + 1 = 0

The equation contains both

\cos^2(\theta)

and

\sin(\theta)

. Substituting

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

allows the equation to be expressed in terms of a single trigonometric function,

\sin(\theta)

.

2

Distribute and simplify the equation into standard quadratic form.

2\sin^2(\theta) + 5\sin(\theta) - 3 = 0

Expanding the equation yields

2 - 2\sin^2(\theta) - 5\sin(\theta) + 1 = 0

, which simplifies to

-2\sin^2(\theta) - 5\sin(\theta) + 3 = 0

. Multiplying the entire equation by

-1

puts it into standard quadratic form

as^2 + bs + c = 0

.

3

Factor the quadratic expression.

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

Finding two numbers that multiply to

-6

(from

2 \times -3

) and add to

5

leads to the factors

6

and

-1

. Splitting the middle term and factoring by grouping yields

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

.

4

Determine the valid solution based on the angle's constraints.

\sin(\theta) = 0.5

Setting each factor to zero gives

\sin(\theta) = 0.5

or

\sin(\theta) = -3

. Since the sine of any real angle must be between

-1

and

1

,

\sin(\theta) = -3

is undefined. Furthermore, because

\theta

is an acute angle (

0^\circ < \theta < 90^\circ

), the sine value must be positive, confirming

\sin(\theta) = 0.5

.

Key Concept

Pythagorean identity and quadratic trigonometric equations

Estimated Time:2m 0s

Question 19Question →

For an acute angle $x$ measured in degrees, $\sin(x) - \sin(90^\circ - x) = \frac{1}{5}$ . What is the value of $12(\tan(x) + \tan(90^\circ - x))$ ?

Show answer & explanation

Answer: 25

Answer

The value of the expression is 25.

Applying the co-function identity

\sin(90^\circ - x) = \cos(x)

allows the given equation to be written as

\sin(x) - \cos(x) = \frac{1}{5}

. Squaring both sides of this equation yields

\sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x) = \frac{1}{25}

. Applying the Pythagorean identity

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

simplifies this to

1 - 2\sin(x)\cos(x) = \frac{1}{25}

, which gives

\sin(x)\cos(x) = \frac{12}{25}

. The expression to be evaluated is

12(\tan(x) + \tan(90^\circ - x))

. Using the identity

\tan(90^\circ - x) = \cot(x)

, this expression can be rewritten as

12\left(\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}\right) = 12\left(\frac{\sin^2(x) + \cos^2(x)}{\sin(x)\cos(x)}\right) = 12\left(\frac{1}{\sin(x)\cos(x)}\right)

. Substituting the value of

\sin(x)\cos(x)

gives

12 \times \frac{25}{12} = 25

.

Step-by-Step Solution

1

Apply the co-function identity to rewrite the equation.

\sin(x) - \cos(x) = \frac{1}{5}

Since

\sin(90^\circ - x) = \cos(x)

for any angle

x

, we can substitute

\cos(x)

into the given equation.

2

Square both sides of the rewritten equation.

\sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x) = \frac{1}{25}

Squaring both sides allows us to use the Pythagorean trigonometric identity to find the product of sine and cosine.

3

Substitute the Pythagorean identity and solve for

\sin(x)\cos(x)

.

\sin(x)\cos(x) = \frac{12}{25}

Substituting

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

yields

1 - 2\sin(x)\cos(x) = \frac{1}{25}

, which simplifies to

\sin(x)\cos(x) = \frac{12}{25}

.

4

Rewrite the target expression in terms of sine and cosine.

\tan(x) + \tan(90^\circ - x) = \frac{1}{\sin(x)\cos(x)}

Using the co-function identity

\tan(90^\circ - x) = \cot(x)

and expressing the tangent and cotangent functions as ratios of sine and cosine yields

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

.

5

Substitute the value of

\sin(x)\cos(x)

and multiply by 12.

25

Substituting

\sin(x)\cos(x) = \frac{12}{25}

into

12(\tan(x) + \tan(90^\circ - x))

gives

12 \times \frac{25}{12} = 25

.

Key Concept

Applying co-function identities, the Pythagorean identity, and fundamental trigonometric relations to simplify expressions.

Estimated Time:2m 0s

Question 20Question →

A right triangle has acute angles $P$ and $R$ . The sine of angle $P$ is defined by $\frac{a}{13}$ and the cosine of angle $R$ is $\frac{5}{13}$ , where $a$ is a positive constant. What is the value of $\sin(R)$ ?

\frac{5}{13}

\frac{12}{13}

\frac{12}{5}

\frac{\sqrt{194}}{13}

Show answer & explanation

Answer: $\frac{12}{13}$

Answer

\frac{12}{13}

The correct answer is the value

\frac{12}{13}

. Since

P

and

R

are the acute angles of a right triangle, they are complementary, which means

\sin(P) = \cos(R) = \frac{5}{13}

. By using the Pythagorean identity

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

, we can substitute the value of

\cos(R)

to get

\sin^2(R) + \left(\frac{5}{13}\right)^2 = 1

. Solving for

\sin(R)

yields

\sin^2(R) = 1 - \frac{25}{169} = \frac{144}{169}

. Since

R

is an acute angle, taking the positive square root yields

\sin(R) = \frac{12}{13}

.

Step-by-Step Solution

1

Apply the complementary angle identity to determine the value of the constant

a

.

Since

P

and

R

are the acute angles of a right triangle, they are complementary, meaning

P + R = 90^\circ

. The co-function identity states that

\sin(P) = \cos(R)

. Given

\cos(R) = \frac{5}{13}

, we have

\sin(P) = \frac{a}{13} = \frac{5}{13}

, which means

a = 5

.

This establishes the relationship between the trigonometric ratios of the two acute angles.

2

Use the Pythagorean identity to find

\sin(R)

.

Using the Pythagorean identity

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

, substitute

\cos(R) = \frac{5}{13}

into the equation:

\sin^2(R) + \left(\frac{5}{13}\right)^2 = 1 \implies \sin^2(R) + \frac{25}{169} = 1 \implies \sin^2(R) = 1 - \frac{25}{169} = \frac{144}{169}

. Taking the positive square root because

R

is an acute angle gives

\sin(R) = \frac{12}{13}

.

This determines the sine of the angle from its known cosine value.

Key Concept

Trigonometric ratios of complementary angles and the Pythagorean identity in right triangles.

Trigonometric Ratios and Identities

Answer

Step-by-Step Solution

Key Concept

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Common Mistakes

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