Question

given right triangle xyz, what is the value of \\(\tan(60^\circ)\\)?

\\(\bigcirc \frac{1}{2}\\)
\\(\bigcirc \frac{\sqrt{3}}{2}\\)
\\(\bigcirc \sqrt{3}\\)
\\(\bigcirc \frac{2}{1}\\)

Explanation:

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<reasoning>

Identify the given values and target ratio

\[

$$\begin{aligned} &\text{Angle } X = 60^\circ\\ &\text{Adjacent side to } 60^\circ \text{ (side } XZ\text{)} = 21\\ &\text{Hypotenuse (side } XY\text{)} = 42 \end{aligned}$$

\]

Calculate the opposite side using the Pythagorean theorem

\[

$$\begin{aligned} &YZ = \sqrt{XY^2 - XZ^2}\\ &YZ = \sqrt{42^2 - 21^2} = \sqrt{1764 - 441} = \sqrt{1323} = 21\sqrt{3} \end{aligned}$$

\]

Compute the tangent of 60 degrees

\[

$$\begin{aligned} &\tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{YZ}{XZ}\\ &\tan(60^\circ) = \frac{21\sqrt{3}}{21} = \sqrt{3} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{2}\)</mcq-option>
<mcq-option>(B) \(\frac{\sqrt{3}}{2}\)</mcq-option>
<mcq-correct>(C) \(\sqrt{3}\)</mcq-correct>
<mcq-option>(D) \(\frac{2}{1}\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the given values and target ratio

\[

$$\begin{aligned} &\text{Angle } X = 60^\circ\\ &\text{Adjacent side to } 60^\circ \text{ (side } XZ\text{)} = 21\\ &\text{Hypotenuse (side } XY\text{)} = 42 \end{aligned}$$

\]

Calculate the opposite side using the Pythagorean theorem

\[

$$\begin{aligned} &YZ = \sqrt{XY^2 - XZ^2}\\ &YZ = \sqrt{42^2 - 21^2} = \sqrt{1764 - 441} = \sqrt{1323} = 21\sqrt{3} \end{aligned}$$

\]

Compute the tangent of 60 degrees

\[

$$\begin{aligned} &\tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{YZ}{XZ}\\ &\tan(60^\circ) = \frac{21\sqrt{3}}{21} = \sqrt{3} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\frac{1}{2}\)</mcq-option>
<mcq-option>(B) \(\frac{\sqrt{3}}{2}\)</mcq-option>
<mcq-correct>(C) \(\sqrt{3}\)</mcq-correct>
<mcq-option>(D) \(\frac{2}{1}\)</mcq-option>
</answer>

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