Question

which equation could be used to solve for the length of xy?
○ $xy = (22)\sin(41^circ)$
○ $xy = (22)\cos(41^circ)$
○ $xy = \frac{22}{\cos(41^circ)}$
○ $xy = \frac{22}{\sin(41^circ)}$

Explanation:

Step1: Identify triangle type

Triangle \( XYZ \) is right - angled at \( Z \). So we can use trigonometric ratios. The trigonometric ratios are: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

Step2: Analyze sides and angle

We know that \( XZ = 22 \) (opposite to angle \( Y = 41^{\circ}\)), and \( XY \) is the hypotenuse. We want to find \( XY \).
From the definition of sine: \(\sin(41^{\circ})=\frac{XZ}{XY}\)
Substituting \( XZ = 22 \) into the formula, we get \(\sin(41^{\circ})=\frac{22}{XY}\)

Step3: Solve for \( XY \)

Cross - multiplying the equation \(\sin(41^{\circ})=\frac{22}{XY}\) gives us \( XY\times\sin(41^{\circ})=22\)
Then, \( XY=\frac{22}{\sin(41^{\circ})}\)

Answer:

\( XY = \frac{22}{\sin(41^{\circ})} \) (the last option)