Question

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

Explanation:

Step1: Use cosine function

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 75^{\circ}\), adjacent side \(PQ = 8\), and hypotenuse \(OQ=x\). So, \(\cos(75^{\circ})=\frac{8}{x}\).

Step2: Solve for \(x\)

Rearrange the formula to \(x=\frac{8}{\cos(75^{\circ})}\). 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:

\(x\approx30.9\)