Question

what is the angle $\theta$ in the triangle below? (hint: use an inverse trigonometric function.)
a. $21.8^{circ}$
b. $0.40^{circ}$
c. $23.6^{circ}$
d. $66.4^{circ}$

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, the cosine of an angle \(\theta\) is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, the adjacent side to the angle \(\theta\) is \(6\) and the hypotenuse is \(15\). So, \(\cos\theta=\frac{6}{15}=\frac{2}{5}\).

Step2: Use the inverse cosine function

We know that if \(\cos\theta = x\), then \(\theta=\cos^{- 1}(x)\). Substituting \(x = \frac{2}{5}\), we get \(\theta=\cos^{-1}(\frac{2}{5})\).
Using a calculator, \(\cos^{-1}(0.4)\approx66.4^{\circ}\)

Answer:

D. \(66.4^{\circ}\)