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:

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

$$ QS = \frac{18}{\sin(Q)} $$

Answer:

• (A) \(18 \cos Q\)
• (B) \(18 \sin Q\)
• (C) \(\frac{18}{\cos Q}\)
• (D) \(\frac{18}{\sin Q}\) (Correct answer)