Question

find the cosine of $\angle d$. simplify your answer and write it as a proper fraction, improper fraction, or whole number. $\cos(d) = $

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 $\angle D$

Adjacent side to $\angle D$: $DE = 39$; Opposite side: $EC = 80$. First calculate hypotenuse $DC$ using Pythagorean theorem:
$$DC = \sqrt{DE^2 + EC^2} = \sqrt{39^2 + 80^2}$$
$$DC = \sqrt{1521 + 6400} = \sqrt{7921} = 89$$

Step3: Compute $\cos(D)$

Substitute adjacent side and hypotenuse into cosine formula:
$$\cos(D) = \frac{DE}{DC} = \frac{39}{89}$$

Answer:

$\frac{39}{89}$