Question

which equation can be used to find the measure of angle bac?
( \tan^{-1}(\frac{5}{12}) = x )
( \tan^{-1}(\frac{12}{5}) = x )
( cos^{-1}(\frac{12}{13}) = x )
( cos^{-1}(\frac{13}{12}) = x )

Explanation:

Step1: Recall trigonometric ratio definitions

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) and \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\) for an acute angle \(\theta\). For angle \(BAC\):

• The side opposite to angle \(BAC\) is \(BC = 12\).
• The side adjacent to angle \(BAC\) is \(AC=5\).
• The hypotenuse is \(AB = 13\).

Step2: Check tangent formula

\(\tan x=\frac{\text{opposite}}{\text{adjacent}}\). If \(x = \angle BAC\), \(\tan x=\frac{BC}{AC}=\frac{12}{5}\). Then, by the inverse - tangent function \(x=\tan^{- 1}(\frac{12}{5})\).

Step3: Check cosine formula

\(\cos x=\frac{\text{adjacent}}{\text{hypotenuse}}\). If \(x=\angle BAC\), \(\cos x=\frac{AC}{AB}=\frac{5}{13}\), not \(\frac{12}{13}\) or \(\frac{13}{12}\) (since \(\frac{13}{12}>1\) and the range of the cosine function is \([-1,1]\)).

Answer:

\(\tan^{-1}(\frac{12}{5}) = x\) (the second option)