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:

Length of side XY:

Step1: Identify triangle type

Triangle XYZ is right - angled at Z, so we can use the Pythagorean theorem. The Pythagorean theorem states that for a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(c^{2}=a^{2}+b^{2}\). Here, \(a = 6\) (length of XZ), \(b = 8\) (length of YZ), and \(c=XY\).

Step2: Apply Pythagorean theorem

\(XY^{2}=XZ^{2}+YZ^{2}\)
\(XY^{2}=6^{2}+8^{2}\)
\(XY^{2}=36 + 64\)
\(XY^{2}=100\)
Taking the square root of both sides, \(XY=\sqrt{100}=10\)

Value of \(\sin(X)\):

Step1: Recall the definition of sine

In a right - triangle, \(\sin(\theta)=\frac{\text{opposite side to } \theta}{\text{hypotenuse}}\). 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 the definition of cosine

In a right - triangle, \(\cos(\theta)=\frac{\text{adjacent side to } \theta}{\text{hypotenuse}}\). 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 side XY: \(10\)
• \(\sin(X)\): \(\frac{4}{5}\)
• \(\cos(X)\): \(\frac{3}{5}\)
• \(\tan(X)\): \(\frac{4}{3}\)