Question

consider △xyz. what are the ratios of sine, cosine, and tangent for angle y? sin(y) = (\frac{xz}{xy}); cos(y) = (\frac{yz}{xy}); tan(y) = (\frac{xz}{yz}) sin(y) = (\frac{yz}{xy}); cos(y) = (\frac{xz}{xy}); tan(y) = (\frac{yz}{xz}) sin(y) = (\frac{xz}{xy}); cos(y) = (\frac{xz}{yz}); tan(y) = (\frac{xy}{yz}) sin(y) = (\frac{yz}{xy}); cos(y) = (\frac{xz}{yz}); tan(y) = (\frac{xy}{xz})

Explanation:

Step1: Recall trigonometric - ratio definitions

In right - triangle $\triangle XYZ$ with right - angle at $Z$, for an acute angle $Y$, $\sin(Y)=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos(Y)=\frac{\text{adjacent}}{\text{hypotenuse}}$, and $\tan(Y)=\frac{\text{opposite}}{\text{adjacent}}$. The side opposite to angle $Y$ is $XZ$, the side adjacent to angle $Y$ is $YZ$, and the hypotenuse is $XY$.

Step2: Calculate the sine, cosine, and tangent of angle $Y$

$\sin(Y)=\frac{XZ}{XY}$ (opposite over hypotenuse), $\cos(Y)=\frac{YZ}{XY}$ (adjacent over hypotenuse), $\tan(Y)=\frac{XZ}{YZ}$ (opposite over adjacent).

Answer:

$\sin(Y)=\frac{XZ}{XY};\cos(Y)=\frac{YZ}{XY};\tan(Y)=\frac{XZ}{YZ}$