Question

from a boat at sea, the measure of the angle of elevation to the top of a lighthouse on the shore is 42°. if the boat is 120 meters from the lighthouse, what is the distance from the boat to the top of the lighthouse? (see yellow hint below.) 80.2 m 179.3 m 161.5 m 89.2 m

Explanation:

Step1: Identify the trigonometric relationship

We know the adjacent side ($120$ m) to the angle of elevation ($42^{\circ}$) and we need to find the hypotenuse (distance from boat to top of lighthouse). Use the cosine function: $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Let $d$ be the distance. So, $\cos42^{\circ}=\frac{120}{d}$.

Step2: Solve for $d$

Rearrange the formula to $d = \frac{120}{\cos42^{\circ}}$. Using a calculator, $\cos42^{\circ}\approx0.7431$. Then $d=\frac{120}{0.7431}\approx161.5$ m.

Answer:

$161.5$ m