Question

solve for x. round to the nearest tenth, if necessary.

Explanation:

Step1: Identify trigonometric ratio

In right triangle $RQS$, $\cos Q = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{RQ}{SQ}$

Step2: Find adjacent side length

$\tan Q = \frac{RS}{RQ} \implies RQ = \frac{RS}{\tan Q} = \frac{35}{\tan 40^\circ}$
$\tan 40^\circ \approx 0.8391$, so $RQ \approx \frac{35}{0.8391} \approx 41.71$

Step3: Calculate hypotenuse $x$

$\cos 40^\circ = \frac{RQ}{x} \implies x = \frac{RQ}{\cos 40^\circ}$
$\cos 40^\circ \approx 0.7660$, so $x \approx \frac{41.71}{0.7660} \approx 43.3$

Alternative direct step:

Step1: Use sine ratio for hypotenuse

$\sin S = \frac{RQ}{x}$, but $\angle S = 90^\circ - 40^\circ = 50^\circ$, so $\sin 50^\circ = \frac{35}{x}$

Step2: Solve for $x$

$x = \frac{35}{\sin 50^\circ}$
$\sin 50^\circ \approx 0.7660$, so $x \approx \frac{35}{0.7660} \approx 43.3$

Answer:

43.3