Question

in the figure below, △abc ~ △xzy. find sin z, tan z, and cos z. round your answers to the nearest hundredth.

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$, and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Since $\triangle ABC\sim\triangle XZY$, the corresponding angles are equal. Angle $Z$ corresponds to angle $B$.

Step2: Calculate $\sin Z$

For angle $B$ in $\triangle ABC$, the opposite side to angle $B$ is $AC = 33$ and the hypotenuse is $AB=43.8$. So, $\sin Z=\sin B=\frac{AC}{AB}=\frac{33}{43.8}\approx0.75$.

Step3: Calculate $\tan Z$

The opposite side to angle $B$ is $AC = 33$ and the adjacent side is $BC = 28.8$. So, $\tan Z=\tan B=\frac{AC}{BC}=\frac{33}{28.8}\approx1.15$.

Step4: Calculate $\cos Z$

The adjacent side to angle $B$ is $BC = 28.8$ and the hypotenuse is $AB = 43.8$. So, $\cos Z=\cos B=\frac{BC}{AB}=\frac{28.8}{43.8}\approx0.66$.

Answer:

$\sin Z\approx0.75$
$\tan Z\approx1.15$
$\cos Z\approx0.66$