Question

which trigonometric ratios are correct for triangle xyz? check all that apply.
□ $\tan(y) = \frac{8}{15}$
□ $\cos(x) = \frac{15}{17}$
□ $\tan(x) = \frac{15}{8}$
□ $\sin(y) = \frac{8}{17}$
□ $\cos(y) = \frac{8}{17}$
(image: right triangle xyz with right angle at z, zy = 15, zx = 8)

Explanation:

Step1: Find hypotenuse XY

Use Pythagorean theorem:
$$XY = \sqrt{YZ^2 + XZ^2} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17$$

Step2: Verify $\tan(Y)$

$\tan(Y)=\frac{\text{opposite}}{\text{adjacent}}=\frac{XZ}{YZ}=\frac{8}{15}$

Step3: Verify $\cos(X)$

$\cos(X)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{XZ}{XY}=\frac{8}{17}
eq \frac{15}{17}$

Step4: Verify $\tan(X)$

$\tan(X)=\frac{\text{opposite}}{\text{adjacent}}=\frac{YZ}{XZ}=\frac{15}{8}$

Step5: Verify $\sin(Y)$

$\sin(Y)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{XZ}{XY}=\frac{8}{17}$

Step6: Verify $\cos(Y)$

$\cos(Y)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{YZ}{XY}=\frac{15}{17}
eq \frac{8}{17}$

Answer:

$\tan(Y) = \frac{8}{15}$, $\tan(X) = \frac{15}{8}$, $\sin(Y) = \frac{8}{17}$