Question

writing inverse trigonometric equations
which equations for the measures of the unknown angles x and y are correct? check all that apply.
$x = \cos^{-1}(\frac{a}{c})$
$x = \sin^{-1}(\frac{c}{b})$
$x = \tan^{-1}(\frac{c}{a})$
$y = \sin^{-1}(\frac{a}{c})$
$y = \cos^{-1}(\frac{c}{b})$

Explanation:

Step1: Recall trigonometric ratios

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

Step2: Analyze for angle \(x\)

For angle \(x\), the opposite side is \(c\) and the adjacent side is \(a\). So, \(\tan x=\frac{c}{a}\), then \(x = \tan^{-1}(\frac{c}{a})\)

Step3: Analyze for angle \(y\)

For angle \(y\), the opposite side is \(a\) and the hypotenuse is \(b\). So, \(\sin y=\frac{a}{b}\) (incorrect). The adjacent side to \(y\) is \(c\) and the hypotenuse is \(b\), so \(\cos y=\frac{c}{b}\), then \(y=\cos^{-1}(\frac{c}{b})\)

Answer:

\(x = \tan^{-1}(\frac{c}{a})\), \(y=\cos^{-1}(\frac{c}{b})\)