Question

what is the length of $overline{ab}$? round to the nearest tenth. c b 10 m 75° x a 9.7 m 10.4 m 37.3 m 38.6 m

Explanation:

Step1: Identify the trigonometric relation

In right - triangle \(ABC\) with right - angle at \(C\), we know the side \(AC = 10\) m and the angle \(A=75^{\circ}\), and we want to find the hypotenuse \(AB=x\). We use the cosine function \(\cos(A)=\frac{AC}{AB}\).
So, \(\cos(75^{\circ})=\frac{10}{x}\).

Step2: Solve for \(x\)

We know that \(\cos(75^{\circ})=\cos(45^{\circ} + 30^{\circ})=\cos45^{\circ}\cos30^{\circ}-\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.259\).
From \(\cos(75^{\circ})=\frac{10}{x}\), we can rewrite it as \(x = \frac{10}{\cos(75^{\circ})}\).
Substitute the value of \(\cos(75^{\circ})\) into the formula: \(x=\frac{10}{0.259}\approx38.6\) m.

Answer:

38.6 m