Question

which equation can be used to find the measure of angle bac?
○ tan⁻¹(5/12)=x
○ tan⁻¹(12/5)=x
○ cos⁻¹(12/13)=x
○ cos⁻¹(13/12)=x

Explanation:

Step1: Recall trigonometric - ratio definitions

In right - triangle \(ABC\) with right - angle at \(C\), for angle \(BAC\), the tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) and the cosine of an angle is defined as \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle BAC\), the opposite side to \(\angle BAC\) is \(BC = 12\), the adjacent side is \(AC = 5\), and the hypotenuse is \(AB=13\).

Step2: Calculate tangent and cosine

The tangent of \(\angle BAC\) is \(\tan(\angle BAC)=\frac{BC}{AC}=\frac{12}{5}\), so \(\angle BAC=\tan^{- 1}(\frac{12}{5})\). The cosine of \(\angle BAC\) is \(\cos(\angle BAC)=\frac{AC}{AB}=\frac{5}{13}\), so \(\angle BAC=\cos^{-1}(\frac{5}{13})\).

Answer:

\(\tan^{-1}(\frac{12}{5}) = x\) (corresponding to the second option in the multiple - choice list)