Question

in triangle abc, what is the approximate measure of angle a? options: 57.1°, 78.5°, 82.4°, 44.4°

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines for finding an angle in a triangle with sides \(a\), \(b\), \(c\) and the angle \(A\) opposite to side \(a\) is \(a^{2}=b^{2}+c^{2}-2bc\cos A\). In \(\triangle ABC\), let \(a = 14\), \(b = 12\), \(c = 10\). We want to find \(\angle A\). So, \(\cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\).

Step2: Substitute the values

Substitute \(b = 12\), \(c = 10\), \(a = 14\) into the formula:
\[

$$\begin{align*} \cos A&=\frac{12^{2}+ 10^{2}-14^{2}}{2\times12\times10}\\ &=\frac{144 + 100-196}{240}\\ &=\frac{48}{240}\\ &=0.2 \end{align*}$$

\]

Step3: Find the angle

Since \(\cos A = 0.2\), then \(A=\cos^{-1}(0.2)\approx78.5^{\circ}\)

Answer:

\(78.5^{\circ}\)