Question

solve for $x$. round to the nearest tenth, if necessary.
answer attempt 1 out of 2
$x =$

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \(OPQ\), we know the adjacent side (\(PQ = 8\)) to the angle \(75^{\circ}\) and we need to find the hypotenuse (\(x=OQ\)). The cosine ratio is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). So, \(\cos(75^{\circ})=\frac{PQ}{OQ}\).

Step2: Solve for \(x\)

Since \(\cos(75^{\circ})=\frac{8}{x}\), we can rewrite the equation as \(x=\frac{8}{\cos(75^{\circ})}\).
We know that \(\cos(75^{\circ})=\cos(45^{\circ}+ 30^{\circ})=\cos45^{\circ}\cos30^{\circ}-\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.259\).
Then \(x = \frac{8}{0.259}\approx30.9\).

Answer:

\(30.9\)