Question

triangle abc is a right triangle, and \\(\sin(53^{\circ}) = \frac{4}{x}\\). solve for x and round to the nearest whole number.

which equation correctly uses the value of x to represent the cosine of angle a?

\\(\cos(53^{\circ}) = \frac{4}{x}\\)
\\(\cos(53^{\circ}) = \frac{y}{5}\\)
\\(\cos(53^{\circ}) = \frac{x}{4}\\)
\\(\cos(53^{\circ}) = \frac{5}{y}\\)

Explanation:

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<reasoning>

Solve for x using the sine ratio

\[

$$\begin{aligned} \sin(53^\circ) &= \frac{4}{x} \\ x &= \frac{4}{\sin(53^\circ)} \\ x &\approx \frac{4}{0.7986} \approx 5.008 \\ x &\approx 5 \end{aligned}$$

\]

Determine the cosine ratio for angle A

\[

$$\begin{aligned} \cos(A) &= \frac{\text{Adjacent}}{\text{Hypotenuse}} \\ \cos(53^\circ) &= \frac{y}{x} \\ \text{Substitute } x = 5: \\ \cos(53^\circ) &= \frac{y}{5} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\cos(53^\circ) = \frac{4}{x}\)</mcq-option>
<mcq-correct>(B) \(\cos(53^\circ) = \frac{y}{5}\)</mcq-correct>
<mcq-option>(C) \(\cos(53^\circ) = \frac{x}{4}\)</mcq-option>
<mcq-option>(D) \(\cos(53^\circ) = \frac{5}{y}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Solve for x using the sine ratio

\[

$$\begin{aligned} \sin(53^\circ) &= \frac{4}{x} \\ x &= \frac{4}{\sin(53^\circ)} \\ x &\approx \frac{4}{0.7986} \approx 5.008 \\ x &\approx 5 \end{aligned}$$

\]

Determine the cosine ratio for angle A

\[

$$\begin{aligned} \cos(A) &= \frac{\text{Adjacent}}{\text{Hypotenuse}} \\ \cos(53^\circ) &= \frac{y}{x} \\ \text{Substitute } x = 5: \\ \cos(53^\circ) &= \frac{y}{5} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\cos(53^\circ) = \frac{4}{x}\)</mcq-option>
<mcq-correct>(B) \(\cos(53^\circ) = \frac{y}{5}\)</mcq-correct>
<mcq-option>(C) \(\cos(53^\circ) = \frac{x}{4}\)</mcq-option>
<mcq-option>(D) \(\cos(53^\circ) = \frac{5}{y}\)</mcq-option>
</answer>

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