Question

find the sine, cosine, and tangent of ∠y. simplify your answers and write them as proper fractions, improper fractions. sin(y) = cos(y) = tan(y) =

Explanation:

Step1: Find the length of side XW

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 26\) (hypotenuse) and \(a = 10\). Let the length of \(XW\) be \(b\). So \(b=\sqrt{26^{2}-10^{2}}=\sqrt{(26 + 10)(26 - 10)}=\sqrt{36\times16}=\sqrt{576}=24\).

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

By the definition of sine in a right - triangle \(\sin(Y)=\frac{\text{opposite}}{\text{hypotenuse}}\). The opposite side to \(\angle Y\) is \(XW = 24\) and the hypotenuse is \(XY=26\). So \(\sin(Y)=\frac{24}{26}=\frac{12}{13}\).

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

By the definition of cosine in a right - triangle \(\cos(Y)=\frac{\text{adjacent}}{\text{hypotenuse}}\). The adjacent side to \(\angle Y\) is \(WY = 10\) and the hypotenuse is \(XY = 26\). So \(\cos(Y)=\frac{10}{26}=\frac{5}{13}\).

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

By the definition of tangent in a right - triangle \(\tan(Y)=\frac{\text{opposite}}{\text{adjacent}}\). The opposite side to \(\angle Y\) is \(XW = 24\) and the adjacent side is \(WY = 10\). So \(\tan(Y)=\frac{24}{10}=\frac{12}{5}\).

Answer:

\(\sin(Y)=\frac{12}{13}\), \(\cos(Y)=\frac{5}{13}\), \(\tan(Y)=\frac{12}{5}\)