In right triangle ABC, the measure of angle B is 90^\circ. If \sin(A) = \frac{5}{13}, what… — SAT

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

$\frac{5}{13}$ Answer
B
$\frac{12}{13}$
C
$\frac{13}{5}$
D
$\frac{8}{13}$

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

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

Since the measure of angle

B

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

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).

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

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