Question

use the law of cosines to find the indicated angle in this triangle. round your answer to the nearest hundredth.

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In our triangle, let the sides be \(a = 7\), \(b=6\), \(c = 10\) and the angle we want to find be \(\theta\) (the angle between sides \(a\) and \(b\)). So we can rearrange the Law of Cosines to solve for \(\cos\theta\): \(\cos\theta=\frac{a^{2}+b^{2}-c^{2}}{2ab}\)

Step2: Substitute the values

Substitute \(a = 7\), \(b = 6\), \(c=10\) into the formula:
\(\cos\theta=\frac{7^{2}+6^{2}-10^{2}}{2\times7\times6}=\frac{49 + 36-100}{84}=\frac{85 - 100}{84}=\frac{- 15}{84}\approx - 0.1786\)

Step3: Find the angle \(\theta\)

To find \(\theta\), we take the arccosine of \(- 0.1786\): \(\theta=\arccos(-0.1786)\)
Using a calculator, \(\theta\approx100.30^{\circ}\) (rounded to the nearest hundredth)

Answer:

\(100.30^{\circ}\)