Question

example 2: calculate the measure of angle bac to the nearest degree.

Explanation:

Step1: Use the cosine - law formula

The cosine - law formula for \(\angle BAC\) (let \(\angle BAC=\alpha\)) in \(\triangle ABC\) is \(BC^{2}=AB^{2}+AC^{2}-2\cdot AB\cdot AC\cdot\cos\alpha\). Here, \(AB = 13\mathrm{cm}\), \(AC = 14\mathrm{cm}\) (assuming the un - labeled side is \(14\mathrm{cm}\) as it is a common Pythagorean triple related side), \(BC = 15\mathrm{cm}\). So, \(15^{2}=13^{2}+14^{2}-2\times13\times14\times\cos\alpha\).

Step2: Simplify the equation

\[

$$\begin{align*} 225&=169 + 196-364\cos\alpha\\ 225&=365-364\cos\alpha\\ 364\cos\alpha&=365 - 225\\ 364\cos\alpha&=140\\ \cos\alpha&=\frac{140}{364}=\frac{5}{13} \end{align*}$$

\]

Step3: Find the angle

\(\alpha=\cos^{-1}(\frac{5}{13})\approx67.38^{\circ}\approx67^{\circ}\)

Answer:

\(67^{\circ}\)