Question

find cosθ, where θ is the angle shown. give an exact value, not a decimal approximation.

Explanation:

Step1: Recall cosine - definition

In a right - triangle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Identify adjacent and hypotenuse

The side adjacent to $\theta$ needs to be found first. Using the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 9$ (hypotenuse) and $b = 4$ (opposite side to $\theta$). Let the adjacent side be $a$. Then $a=\sqrt{c^{2}-b^{2}}=\sqrt{9^{2}-4^{2}}=\sqrt{81 - 16}=\sqrt{65}$.

Step3: Calculate cosine value

Since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and the adjacent side is $\sqrt{65}$ and the hypotenuse is 9, so $\cos\theta=\frac{\sqrt{65}}{9}$.

Answer:

$\frac{\sqrt{65}}{9}$