Question

use the law of cosines to find the indicated angle in this triangle.
4
?°
7
5
round your answer to the nearest hundredth.

Explanation:

Step1: Recall Law of Cosines

For a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite side \(c\), the law of cosines is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). Here, the side opposite the unknown angle is \(7\), and the other two sides are \(4\) and \(5\). So let \(a = 4\), \(b = 5\), \(c=7\). Then:

$$7^{2}=4^{2}+5^{2}-2\times4\times5\times\cos\theta$$

Step2: Simplify the Equation

Calculate the squares: \(49 = 16 + 25-40\cos\theta\). Then simplify the right - hand side: \(49=41 - 40\cos\theta\).

Step3: Solve for \(\cos\theta\)

Subtract \(41\) from both sides: \(49 - 41=-40\cos\theta\), so \(8=-40\cos\theta\). Then divide both sides by \(- 40\): \(\cos\theta=\frac{8}{-40}=-0.2\).

Step4: Find \(\theta\)

Take the arccosine of \(-0.2\): \(\theta=\arccos(-0.2)\). Using a calculator, \(\theta\approx101.54^{\circ}\) (rounded to the nearest hundredth).

Answer:

\(101.54^{\circ}\)