Question

writing trigonometric ratios. use the diagram of triangle xyz to answer the questions. what is the length of side xy? what is the value of sin(x)? 5 10 50 100 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 \(a^{2}+b^{2}=c^{2}\), where \(a = 6\), \(b = 8\) and \(c=XY\). So \(XY=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\).

Step2: Define sine ratio

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

Step3: Define cosine ratio

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

Step4: Define tangent ratio

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

Answer:

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