Question

what is the length of $overline{ab}$? round to the nearest tenth.
9.7 m
10.4 m
37.3 m
38.6 m

Explanation:

Step1: Identify the trigonometric relation

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

Step2: Rearrange the formula to solve for $AB$

From $\cos(A)=\frac{AC}{AB}$, we can rewrite it as $AB=\frac{AC}{\cos(A)}$.

Step3: Substitute the values

Given $A = 75^{\circ}$ and $AC = 10$m. $\cos(75^{\circ})=\cos(45^{\circ}+ 30^{\circ})=\cos45^{\circ}\cos30^{\circ}-\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}-\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\approx0.259$. Then $AB=\frac{10}{\cos(75^{\circ})}=\frac{10}{0.259}\approx38.6$m.

Answer:

38.6 m