Question

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

Explanation:

Step1: Find the length of side XY

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 34\) and \(a = 30\). Let \(XY=b\). Then \(b=\sqrt{34^{2}-30^{2}}=\sqrt{(34 + 30)(34 - 30)}=\sqrt{64\times4}=\sqrt{256}=16\).

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

The definition of sine in a right - triangle is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle Y\), the opposite side to \(\angle Y\) is \(XZ = 30\) and the hypotenuse is \(YZ=34\). So \(\sin(Y)=\frac{30}{34}=\frac{15}{17}\).

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

The definition of cosine in a right - triangle is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle Y\), the adjacent side to \(\angle Y\) is \(XY = 16\) and the hypotenuse is \(YZ = 34\). So \(\cos(Y)=\frac{16}{34}=\frac{8}{17}\).

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

The definition of tangent in a right - triangle is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\angle Y\), the opposite side to \(\angle Y\) is \(XZ = 30\) and the adjacent side is \(XY = 16\). So \(\tan(Y)=\frac{30}{16}=\frac{15}{8}\).

Answer:

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