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)?
what is the value of cos(x)?
what is the value of tan(x)?
(diagram: right triangle xyz with right angle at z, yz = 8, xz = 6)

Explanation:

Step1: Calculate hypotenuse XY

Use Pythagorean theorem: $XY = \sqrt{YZ^2 + XZ^2} = \sqrt{8^2 + 6^2}$

Step2: Simplify hypotenuse length

$\sqrt{64 + 36} = \sqrt{100} = 10$

Step3: Find $\sin(X)$

Opposite over hypotenuse: $\sin(X) = \frac{YZ}{XY} = \frac{8}{10} = \frac{4}{5}$

Step4: Find $\cos(X)$

Adjacent over hypotenuse: $\cos(X) = \frac{XZ}{XY} = \frac{6}{10} = \frac{3}{5}$

Step5: Find $\tan(X)$

Opposite over adjacent: $\tan(X) = \frac{YZ}{XZ} = \frac{8}{6} = \frac{4}{3}$

Answer:

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