Question

find the cosine of $\angle u$.
write your answer in simplified, rationalized form. do not round.
$\cos (u) = \square$

Explanation:

Step1: Recall cosine definition for right triangles

For an acute angle in a right triangle, $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$

Step2: Identify sides for $\angle U$

Adjacent side to $\angle U$: $UV = 2$
First, calculate hypotenuse $UW$ using Pythagorean theorem:
$$UW = \sqrt{UV^2 + VW^2} = \sqrt{2^2 + (\sqrt{15})^2} = \sqrt{4 + 15} = \sqrt{19}$$

Step3: Compute $\cos(U)$

Substitute values into cosine formula:
$$\cos(U) = \frac{2}{\sqrt{19}}$$
Rationalize the denominator:
$$\cos(U) = \frac{2\sqrt{19}}{19}$$

Answer:

$\frac{2\sqrt{19}}{19}$