Question

use the law of cosines to find the indicated side length in this triangle. ? 4 29° 10 round your answer to the nearest hundredth.

Explanation:

Step1: Recall Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos(\theta)\). Here, let the side we need to find be \(x\), \(a = 10\), \(b=4\), and the included angle \(\theta = 29^{\circ}\). So the formula for \(x\) is \(x^{2}=10^{2}+4^{2}-2\times10\times4\times\cos(29^{\circ})\).

Step2: Calculate each term

First, calculate \(10^{2}=100\), \(4^{2} = 16\), \(2\times10\times4=80\). Then, find \(\cos(29^{\circ})\approx0.8746\).
Now substitute into the formula: \(x^{2}=100 + 16-80\times0.8746\).
Calculate \(100 + 16=116\), \(80\times0.8746 = 69.968\). Then \(x^{2}=116 - 69.968=46.032\).

Step3: Find \(x\)

Take the square root of \(46.032\): \(x=\sqrt{46.032}\approx6.79\).

Answer:

\(6.79\)