Question

which of the following is correct for finding the measure of angle z?
∠z = \sin^{-1}48/55
∠z = \tan^{-1}55/48
∠z = \cos^{-1}55/73
∠z = \tan^{-1}48/73

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)

For \(\angle Z\) in the right - triangle \(\triangle XYZ\) with right - angle at \(Y\):

• The side opposite to \(\angle Z\) is \(XY = 48\)
• The side adjacent to \(\angle Z\) is \(YZ=55\)
• The hypotenuse is \(XZ = 73\)

Step2: Check each option

• Option 1 (\(\angle Z=\sin^{- 1}\frac{48}{55}\)):

Since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), for \(\angle Z\), \(\sin Z=\frac{XY}{XZ}=\frac{48}{73}
eq\frac{48}{55}\)

• Option 2 (\(\angle Z=\tan^{-1}\frac{55}{48}\)):

Since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for \(\angle Z\), \(\tan Z=\frac{XY}{YZ}=\frac{48}{55}
eq\frac{55}{48}\)

• Option 3 (\(\angle Z=\cos^{-1}\frac{55}{73}\)):

Since \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), for \(\angle Z\), \(\cos Z=\frac{YZ}{XZ}=\frac{55}{73}\)

• Option 4 (\(\angle Z=\tan^{-1}\frac{48}{73}\)):

Since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), for \(\angle Z\), \(\tan Z=\frac{XY}{YZ}=\frac{48}{55}
eq\frac{48}{73}\)

Answer:

\(\angle Z=\cos^{-1}\frac{55}{73}\) (the orange - colored option)