Question

the law of cosines is used to find the measure of ∠z. in the nearest whole degree, what is the measure of ∠z? 41° 47° 51° 57°

Explanation:

Step1: Recall the law of cosines formula

For a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\), \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In \(\triangle XYZ\), let \(a = 18\), \(b = 18\), \(c = 15\), and the angle we want to find is \(\angle Z\). So, \(15^{2}=18^{2}+18^{2}-2\times18\times18\times\cos Z\).

Step2: Simplify the equation

\[

$$\begin{align*} 225&=324 + 324- 648\cos Z\\ 225&=648-648\cos Z\\ 648\cos Z&=648 - 225\\ 648\cos Z&=423\\ \cos Z&=\frac{423}{648}\approx0.6528 \end{align*}$$

\]

Step3: Find the angle

\(Z=\cos^{-1}(0.6528)\approx 49^{\circ}\)

Answer:

\(49^{\circ}\)