Question

find the sine, cosine, and tangent of $\angle y$. simplify your answers and write them as proper fractions, improper fractions, or whole numbers. $\sin(y) = \square$ $\cos(y) = \square$ $\tan(y) = \square$

Explanation:

Step1: Find the length of XY

In right triangle \( XYW \), by Pythagorean theorem \( XY^{2}+XW^{2}=YW^{2} \). Given \( YW = 68 \), \( XW=60 \), so \( XY=\sqrt{YW^{2}-XW^{2}}=\sqrt{68^{2}-60^{2}}=\sqrt{(68 - 60)(68 + 60)}=\sqrt{8\times128}=\sqrt{1024}=32 \).

Step2: Calculate \( \sin(Y) \)

\( \sin(Y)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{XW}{YW}=\frac{60}{68}=\frac{15}{17} \).

Step3: Calculate \( \cos(Y) \)

\( \cos(Y)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{XY}{YW}=\frac{32}{68}=\frac{8}{17} \).

Step4: Calculate \( \tan(Y) \)

\( \tan(Y)=\frac{\text{opposite}}{\text{adjacent}}=\frac{XW}{XY}=\frac{60}{32}=\frac{15}{8} \).

Answer:

\( \sin(Y)=\frac{15}{17} \)
\( \cos(Y)=\frac{8}{17} \)
\( \tan(Y)=\frac{15}{8} \)