Question

find the value of each trigonometric ratio. express answers as simplified fractions.

1. \\(cos z\\)

\\(\

$$\begin{tikzpicture}scale=0.5 \\coordinate (z) at (0,0); \\coordinate (y) at (0,3); \\coordinate (x) at (3,3); \\draw (z) -- (y) -- (x) -- (z); \\draw right angle mark=(y,z,x); \ ode at (z) below left {$z$}; \ ode at (y) above {$y$}; \ ode at (x) below right {$x$}; \ ode at (midway between z and y) left {$36$}; \ ode at (midway between y and x) above {$15$}; \ ode at (midway between z and x) below {$39$}; \\end{tikzpicture}$$

Explanation:

Step1: Recall cosine definition

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

Step2: Identify sides for angle Z

Adjacent to $\angle Z$: $YZ = 36$, Hypotenuse: $ZX = 39$.

Step3: Substitute and simplify fraction

$\cos Z = \frac{36}{39} = \frac{12}{13}$ (divide numerator/denominator by 3).

Answer:

$\frac{12}{13}$