Question

use the triangle shown to the right to evaluate the following expression. if necessary, express the value without a square root in the denominator by rationalizing the denominator. cos 45°

Explanation:

Step1: Recall cosine - ratio definition

The cosine of an angle in a right - triangle is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.
For $\theta = 45^{\circ}$ in the given right - triangle, the adjacent side to the $45^{\circ}$ angle has length $1$ and the hypotenuse has length $\sqrt{2}$. So, $\cos45^{\circ}=\frac{1}{\sqrt{2}}$.

Step2: Rationalize the denominator

To rationalize the denominator of $\frac{1}{\sqrt{2}}$, we multiply the numerator and denominator by $\sqrt{2}$. We get $\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{2}}{2}$.

Answer:

$\frac{\sqrt{2}}{2}$