Question

19
note: figure not drawn to scale.
in triangle qrs shown, \\(qr < rs\\). which expression represents the length of \\(\overline{qs}\\)?
a) \\(18 \cos q\\)
b) \\(18 \sin q\\)
c) \\(\frac{18}{\cos q}\\)
d) \\(\frac{18}{\sin q}\\)

Explanation:

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<reasoning>

Identify the given values and target

In right triangle \(QRS\), the right angle is at vertex \(R\).
The side opposite to angle \(Q\) is \(RS = 18\).
The hypotenuse is \(QS\).

Apply the trigonometric ratio

\[
\sin(Q) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{RS}{QS}
\]
\[
\sin(Q) = \frac{18}{QS}
\]

Solve for the length of QS

\[
QS = \frac{18}{\sin(Q)}
\]
</reasoning>

<answer>
<mcq-option>(A) \(18 \cos Q\)</mcq-option>
<mcq-option>(B) \(18 \sin Q\)</mcq-option>
<mcq-option>(C) \(\frac{18}{\cos Q}\)</mcq-option>
<mcq-correct>(D) \(\frac{18}{\sin Q}\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Identify the given values and target

In right triangle \(QRS\), the right angle is at vertex \(R\).
The side opposite to angle \(Q\) is \(RS = 18\).
The hypotenuse is \(QS\).

Apply the trigonometric ratio

\[
\sin(Q) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{RS}{QS}
\]
\[
\sin(Q) = \frac{18}{QS}
\]

Solve for the length of QS

\[
QS = \frac{18}{\sin(Q)}
\]
</reasoning>

<answer>
<mcq-option>(A) \(18 \cos Q\)</mcq-option>
<mcq-option>(B) \(18 \sin Q\)</mcq-option>
<mcq-option>(C) \(\frac{18}{\cos Q}\)</mcq-option>
<mcq-correct>(D) \(\frac{18}{\sin Q}\)</mcq-correct>
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