Question

1. find the measure of angle xyz to the nearest degree. show your calculations clearly for full marks.

Explanation:

Step1: Use the cosine - law formula

The cosine - law for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In \(\triangle XYZ\), let \(a = 8\mathrm{cm}\), \(b = 6.5\mathrm{cm}\), \(c = 6\mathrm{cm}\), and we want to find \(\angle XYZ=\theta\). Then \(\cos\theta=\frac{a^{2}+b^{2}-c^{2}}{2ab}\).

Step2: Substitute the values of \(a\), \(b\), and \(c\)

\[

$$\begin{align*} \cos\theta&=\frac{8^{2}+6.5^{2}-6^{2}}{2\times8\times6.5}\\ &=\frac{64 + 42.25-36}{104}\\ &=\frac{70.25}{104}\\ &\approx0.6755 \end{align*}$$

\]

Step3: Find the angle

\(\theta=\cos^{- 1}(0.6755)\approx47^{\circ}\)

Answer:

\(47^{\circ}\)