Question

using the law of cosines, what is ( mangle x ) to the nearest degree? ( 61^{circ} ) ( 79^{circ} ) ( 82^{circ} ) ( 91^{circ} )

Explanation:

Step1: Recall the law of cosines formula

The law of cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\). In \(\triangle XYZ\), if we want to find \(\angle X\), let \(a = YZ=9\), \(b = XZ = 8\), \(c=XY = 6\). Then \(9^{2}=8^{2}+6^{2}-2\times8\times6\times\cos X\).

Step2: Simplify the equation

First, calculate the squares: \(81=64 + 36-96\cos X\). Then \(81=100-96\cos X\).

Step3: Solve for \(\cos X\)

Rearrange the equation: \(96\cos X=100 - 81\). So \(96\cos X=19\), and \(\cos X=\frac{19}{96}\approx0.1979\).

Step4: Find the angle \(X\)

Using the inverse - cosine function \(X=\cos^{-1}(0.1979)\). Using a calculator, \(X\approx82^{\circ}\).

Answer:

\(82^{\circ}\)