Question

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

Explanation:

Step1: Recall trigonometric - ratio definitions

In right - triangle \(ABC\) with right - angle at \(C\), for angle \(A\), \(\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}\) and \(\tan A=\frac{\text{opposite}}{\text{adjacent}}\). The side adjacent to angle \(A\) is \(AC = 5\), the side opposite to angle \(A\) is \(BC = 12\), and the hypotenuse \(AB=13\).

Step2: Determine the correct inverse - trigonometric function

To find angle \(A\), if we use the tangent function, \(\tan A=\frac{BC}{AC}=\frac{12}{5}\), then \(A = \tan^{- 1}(\frac{12}{5})\). If we use the cosine function, \(\cos A=\frac{AC}{AB}=\frac{5}{13}\), then \(A=\cos^{-1}(\frac{5}{13})\).

Answer:

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