Question

in the figure below, △xyz ~ △cab. find tan a, sin a, and cos a. round your answers to the nearest hundredth.

Explanation:

Step1: Use similarity of triangles

Since $\triangle XYZ\sim\triangle CAB$, the ratios of corresponding sides are equal. Let's assume the side opposite to angle $A$ in $\triangle CAB$ corresponds to side $YZ$ in $\triangle XYZ$ and the side adjacent to angle $A$ in $\triangle CAB$ corresponds to side $AC$ in $\triangle XYZ$. The hypotenuse of $\triangle XYZ$ is $YZ = 42.5$, the side opposite to the right - angle in $\triangle XYZ$ is $XY=38.5$ and the other side is $XZ = 18$.

Step2: Recall trigonometric ratio definitions

The tangent of an angle in a right - triangle is $\tan A=\frac{\text{opposite}}{\text{adjacent}}$. For similar triangles, the ratio of corresponding sides gives $\tan A=\frac{38.5}{18}\approx2.14$.

Step3: Calculate sine

The sine of an angle in a right - triangle is $\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$. So $\sin A=\frac{38.5}{42.5}\approx0.91$.

Step4: Calculate cosine

The cosine of an angle in a right - triangle is $\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}$. So $\cos A=\frac{18}{42.5}\approx0.42$.

Answer:

$\tan A\approx2.14$
$\sin A\approx0.91$
$\cos A\approx0.42$