Question

which trigonometric ratios are correct for triangle xyz? choose three correct answers.
tan (y)=\frac{8}{15}
cos (y)=\frac{15}{17}
sin (y)=\frac{8}{17}

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For $\angle Y$ in right - triangle $XYZ$ with legs $8$ and $15$, first find the hypotenuse using the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}=\sqrt{8^{2}+15^{2}}=\sqrt{64 + 225}=\sqrt{289}=17$.

Step2: Calculate $\tan(Y)$

$\tan(Y)=\frac{XZ}{YZ}$. Given $XZ = 8$ and $YZ = 15$, so $\tan(Y)=\frac{8}{15}$.

Step3: Calculate $\cos(Y)$

$\cos(Y)=\frac{YZ}{XY}$. Since $YZ = 15$ and $XY = 17$, $\cos(Y)=\frac{15}{17}$.

Step4: Calculate $\sin(Y)$

$\sin(Y)=\frac{XZ}{XY}$. As $XZ = 8$ and $XY = 17$, $\sin(Y)=\frac{8}{17}$.

Answer:

$\tan(Y)=\frac{8}{15}$, $\sin(Y)=\frac{8}{17}$, $\cos(Y)=\frac{15}{17}$ (The first and third options are correct. Assuming the second option has a typo and should be $\cos(Y)=\frac{15}{17}$)