Question

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

Explanation:

Step1: Identify triangle type and trigonometric ratios

The triangle \( \triangle XYZ \) is a right - triangle with \( \angle Z = 90^{\circ} \), \( XZ = 22 \), and \( \angle Y=41^{\circ} \). We know that in a right - triangle, the cosine of an angle \( \theta \) is given by \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \), the sine of an angle \( \theta \) is given by \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \), and the secant of an angle \( \theta \) is \( \sec\theta=\frac{1}{\cos\theta}=\frac{\text{hypotenuse}}{\text{adjacent}} \), the cosecant of an angle \( \theta \) is \( \csc\theta=\frac{1}{\sin\theta}=\frac{\text{hypotenuse}}{\text{opposite}} \).

For \( \angle Y = 41^{\circ} \), the side adjacent to \( \angle Y \) is \( YZ \), the side opposite to \( \angle Y \) is \( XZ = 22 \), and the hypotenuse is \( XY \).

We know that \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \), so \( \sin(41^{\circ})=\frac{XZ}{XY} \). Since \( XZ = 22 \), we can re - arrange the formula for \( XY \): \( XY=\frac{XZ}{\sin(41^{\circ})}=\frac{22}{\sin(41^{\circ})} \).

Let's check the other options:

• For \( XY=(22)\sin(41^{\circ}) \): \( \sin(41^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{XZ}{XY} \), so \( XY\sin(41^{\circ}) = XZ \), \( XY=\frac{XZ}{\sin(41^{\circ})}

eq22\sin(41^{\circ}) \) (unless \( XZ = 22\sin^{2}(41^{\circ}) \), which is not the case here).

• For \( XY=(22)\cos(41^{\circ}) \): \( \cos(41^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{YZ}{XY} \), so \( XY\cos(41^{\circ})=YZ

eq XZ \), so this is incorrect.

• For \( XY = \frac{22}{\cos(41^{\circ})} \): \( \cos(41^{\circ})=\frac{YZ}{XY} \), so \( XY=\frac{YZ}{\cos(41^{\circ})}

eq\frac{22}{\cos(41^{\circ})} \) (since \( YZ
eq22 \) in general for this triangle).

Answer:

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