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)? 5 10 50 100

Explanation:

Length of side \( XY \)

Step1: Identify triangle type

Triangle \( XYZ \) is right - angled at \( Z \), so we can use the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( a = 6 \), \( b = 8 \), and \( c=XY \).

Step2: Apply Pythagorean theorem

\( XY^{2}=XZ^{2}+YZ^{2}=6^{2}+8^{2}=36 + 64=100 \)

Step3: Solve for \( XY \)

Take the square root of both sides: \( XY=\sqrt{100} = 10 \)

Value of \( \sin(X) \)

Step1: Recall sine definition

In a right - triangle, \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle X \), the opposite side is \( YZ = 8 \) and the hypotenuse is \( XY = 10 \).

Step2: Calculate \( \sin(X) \)

\( \sin(X)=\frac{YZ}{XY}=\frac{8}{10}=\frac{4}{5} \)

Value of \( \cos(X) \)

Step1: Recall cosine definition

In a right - triangle, \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). For \( \angle X \), the adjacent side is \( XZ = 6 \) and the hypotenuse is \( XY = 10 \).

Step2: Calculate \( \cos(X) \)

\( \cos(X)=\frac{XZ}{XY}=\frac{6}{10}=\frac{3}{5} \)

Value of \( \tan(X) \)

Answer:

s:

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