Question

use the diagram of triangle xyz to answer the questions. what is the length of side xy? what is the value of sin(x)? what is the value of cos(x)? what is the value of tan(x)?

Explanation:

Step1: Find length of XY using Pythagorean theorem

In right - triangle XYZ, by the Pythagorean theorem \(XY^{2}=XZ^{2}+YZ^{2}\). Given \(XZ = 6\) and \(YZ = 8\), then \(XY=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\).

Step2: Find \(\sin(X)\)

The definition of sine in a right - triangle is \(\sin(X)=\frac{\text{opposite}}{\text{hypotenuse}}\). The side opposite to angle \(X\) is \(YZ\) and the hypotenuse is \(XY\). So \(\sin(X)=\frac{YZ}{XY}=\frac{8}{10}=\frac{4}{5}\).

Step3: Find \(\cos(X)\)

The definition of cosine in a right - triangle is \(\cos(X)=\frac{\text{adjacent}}{\text{hypotenuse}}\). The side adjacent to angle \(X\) is \(XZ\) and the hypotenuse is \(XY\). So \(\cos(X)=\frac{XZ}{XY}=\frac{6}{10}=\frac{3}{5}\).

Step4: Find \(\tan(X)\)

The definition of tangent in a right - triangle is \(\tan(X)=\frac{\text{opposite}}{\text{adjacent}}\). The side opposite to angle \(X\) is \(YZ\) and the side adjacent to angle \(X\) is \(XZ\). So \(\tan(X)=\frac{YZ}{XZ}=\frac{8}{6}=\frac{4}{3}\).

Answer:

Length of \(XY\): 10
\(\sin(X)\): \(\frac{4}{5}\)
\(\cos(X)\): \(\frac{3}{5}\)
\(\tan(X)\): \(\frac{4}{3}\)