In right triangle CDE, the right angle is located at vertex D. The length of leg CD is 24 … — ACT

In right triangle $CDE$ , the right angle is located at vertex $D$ . The length of leg $CD$ is $24$ centimeters. If $\tan(E) = \frac{4}{3}$ , what is the value of $\sin(C)$ ?

$\frac{3}{5}$ Answer
B
$\frac{4}{5}$
C
$\frac{3}{4}$
D
$\frac{4}{3}$
E
$\frac{5}{3}$

Answer

The value of

\sin(C)

\frac{3}{5}

To find

\sin(C)

, first express

\tan(E) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{DE} = \frac{24}{DE} = \frac{4}{3}

, which gives

DE = 18

. Next, compute the hypotenuse

CE = \sqrt{24^2 + 18^2} = \sqrt{900} = 30

. Finally, identify the side opposite to angle

C

, which is

DE = 18

. Thus,

\sin(C) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{18}{30} = \frac{3}{5}

. The option stating

\frac{3}{5}

is correct.

Step-by-Step Solution

Use the definition of tangent for angle

E

to find the length of side

DE

DE = 18\text{ cm}

Since

\tan(E) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{DE}

, substituting

CD = 24

yields

\frac{24}{DE} = \frac{4}{3}

, so

DE = \frac{24 \times 3}{4} = 18

Calculate the hypotenuse

CE

using the Pythagorean theorem.

CE = 30\text{ cm}

In right triangle

CDE

CE^2 = CD^2 + DE^2 = 24^2 + 18^2 = 576 + 324 = 900

, so

CE = \sqrt{900} = 30

Calculate

\sin(C)

using the SOHCAHTOA ratio for angle

C

\sin(C) = \frac{3}{5}

Relative to angle

C

, the opposite side is

DE = 18

and the hypotenuse is

CE = 30

. Thus,

\sin(C) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{18}{30} = \frac{3}{5}

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)

Estimated Time:1m 15s

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