Question

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

Explanation:

Step1: Recall Trigonometric Ratios

In a right - triangle, for an acute angle \(\theta\):

• \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)
• \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
• \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

In \(\triangle XYZ\) with right - angle at \(Z\), for angle \(Y\):

• The side opposite to angle \(Y\) is \(XZ\).
• The side adjacent to angle \(Y\) is \(YZ\).
• The hypotenuse (the side opposite the right - angle) is \(XY\).

Step2: Calculate \(\sin(Y)\), \(\cos(Y)\) and \(\tan(Y)\)

• For \(\sin(Y)\): Using the formula \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), we have \(\sin(Y)=\frac{\text{opposite to }Y}{\text{hypotenuse}}=\frac{XZ}{XY}\)
• For \(\cos(Y)\): Using the formula \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), we have \(\cos(Y)=\frac{\text{adjacent to }Y}{\text{hypotenuse}}=\frac{YZ}{XY}\)
• For \(\tan(Y)\): Using the formula \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), we have \(\tan(Y)=\frac{\text{opposite to }Y}{\text{adjacent to }Y}=\frac{XZ}{YZ}\) (Wait, no, wait. Wait, opposite to \(Y\) is \(XZ\), adjacent to \(Y\) is \(YZ\), so \(\tan(Y)=\frac{XZ}{YZ}\)? Wait, no, let's re - check. Wait, in right - triangle \(XYZ\), right - angled at \(Z\). So sides: \(XZ\) and \(YZ\) are legs, \(XY\) is hypotenuse. Angle at \(Y\): the side opposite angle \(Y\) is \(XZ\), the side adjacent to angle \(Y\) is \(YZ\). So \(\sin(Y)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{XZ}{XY}\), \(\cos(Y)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{YZ}{XY}\), \(\tan(Y)=\frac{\text{opposite}}{\text{adjacent}}=\frac{XZ}{YZ}\). Let's check the options:

Looking at the options:

Option 1: \(\sin(Y)=\frac{XZ}{XY}\); \(\cos(Y)=\frac{YZ}{XY}\); \(\tan(Y)=\frac{YZ}{XY}\) → No, \(\tan\) is wrong.

Option 2: \(\sin(Y)=\frac{XY}{XZ}\); \(\cos(Y)=\frac{XZ}{XY}\); \(\tan(Y)=\frac{YZ}{XZ}\) → No.

Option 3: \(\sin(Y)=\frac{XZ}{XY}\); \(\cos(Y)=\frac{YZ}{XY}\); \(\tan(Y)=\frac{XZ}{YZ}\) → This matches our calculation.

Option 4: \(\sin(Y)=\frac{YZ}{XY}\); \(\cos(Y)=\frac{XZ}{XY}\); \(\tan(Y)=\frac{XZ}{YZ}\) → \(\sin\) is wrong.

Answer:

The correct option is \(\boldsymbol{\sin(Y)=\frac{XZ}{XY};\cos(Y)=\frac{YZ}{XY};\tan(Y)=\frac{XZ}{YZ}}\) (the third option among the given options).