Question

which trigonometric ratios are correct for triangle xyz? check all that apply.
□ tan(y) = 8/15
□ cos(x) = 15/17
□ tan(x) = 8/15
□ sin(y) = 8/17
□ cos(y) = 8/17

Explanation:

Step1: Recall trigonometric - ratio formulas

For a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In right - triangle $XYZ$ with right - angle at $Z$, by the Pythagorean theorem, the hypotenuse $XY=\sqrt{8^{2}+15^{2}}=\sqrt{64 + 225}=\sqrt{289}=17$.

Step2: Calculate $\sin(Y)$

For $\angle Y$, the opposite side to $\angle Y$ is $XZ = 8$ and the hypotenuse is $XY=17$. So, $\sin(Y)=\frac{XZ}{XY}=\frac{8}{17}$.

Step3: Calculate $\cos(Y)$

For $\angle Y$, the adjacent side to $\angle Y$ is $YZ = 15$ and the hypotenuse is $XY = 17$. So, $\cos(Y)=\frac{YZ}{XY}=\frac{15}{17}$.

Step4: Calculate $\tan(Y)$

For $\angle Y$, the opposite side is $XZ = 8$ and the adjacent side is $YZ = 15$. So, $\tan(Y)=\frac{XZ}{YZ}=\frac{8}{15}$.

Step5: Calculate $\sin(X)$

For $\angle X$, the opposite side to $\angle X$ is $YZ = 15$ and the hypotenuse is $XY=17$. So, $\sin(X)=\frac{YZ}{XY}=\frac{15}{17}$.

Step6: Calculate $\cos(X)$

For $\angle X$, the adjacent side to $\angle X$ is $XZ = 8$ and the hypotenuse is $XY = 17$. So, $\cos(X)=\frac{XZ}{XY}=\frac{8}{17}$.

Step7: Calculate $\tan(X)$

For $\angle X$, the opposite side is $YZ = 15$ and the adjacent side is $XZ = 8$. So, $\tan(X)=\frac{YZ}{XZ}=\frac{15}{8}$.

Answer:

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