Question

given right triangle def, what is the value of \\(\sin(e)\\)?

\\(\frac{4}{3}\\)

\\(\frac{3}{4}\\)

\\(\frac{3}{5}\\)

\\(\frac{4}{5}\\)

Explanation:

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<reasoning>

Find the missing side length DF

\[

$$\begin{aligned} &ED^2 + DF^2 = EF^2\\ &8^2 + DF^2 = 10^2\\ &64 + DF^2 = 100\\ &DF^2 = 36\\ &DF = 6 \end{aligned}$$

\]

Calculate the sine of angle E

\[

$$\begin{aligned} &\sin(E) = \frac{\text{Opposite}}{\text{Hypotenuse}}\\ &\sin(E) = \frac{DF}{EF}\\ &\sin(E) = \frac{6}{10} = \frac{3}{5} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{4}{3}\)</mcq-option>
<mcq-option>(B) \(\frac{3}{4}\)</mcq-option>
<mcq-correct>(C) \(\frac{3}{5}\)</mcq-correct>
<mcq-option>(D) \(\frac{4}{5}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Find the missing side length DF

\[

$$\begin{aligned} &ED^2 + DF^2 = EF^2\\ &8^2 + DF^2 = 10^2\\ &64 + DF^2 = 100\\ &DF^2 = 36\\ &DF = 6 \end{aligned}$$

\]

Calculate the sine of angle E

\[

$$\begin{aligned} &\sin(E) = \frac{\text{Opposite}}{\text{Hypotenuse}}\\ &\sin(E) = \frac{DF}{EF}\\ &\sin(E) = \frac{6}{10} = \frac{3}{5} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{4}{3}\)</mcq-option>
<mcq-option>(B) \(\frac{3}{4}\)</mcq-option>
<mcq-correct>(C) \(\frac{3}{5}\)</mcq-correct>
<mcq-option>(D) \(\frac{4}{5}\)</mcq-option>
</answer>

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