Question

use the law of cosines to solve for the indicated values. round decimal answers to the nearest hundredth.
a) ( mangle a=) type your answer...

Explanation:

Step1: Recall the Law of Cosines formula

The Law of Cosines formula is \(b^{2}=a^{2}+c^{2}-2ac\cos B\). In \(\triangle ABC\), if we want to find \(\angle A\), we use the formula \(BC^{2}=AB^{2}+AC^{2}-2(AB)(AC)\cos A\). Let \(AB = 12\), \(AC=13\), \(BC = 18\). Then \(18^{2}=12^{2}+13^{2}-2\times12\times13\times\cos A\).

Step2: Simplify the equation

First, calculate the squares: \(324=144 + 169-312\cos A\). Then \(324=313-312\cos A\).

Step3: Solve for \(\cos A\)

Rearrange the equation: \(312\cos A=313 - 324\). So \(312\cos A=- 11\). Then \(\cos A=\frac{-11}{312}\approx - 0.035256\).

Step4: Find the angle \(A\)

Since \(A=\cos^{-1}(\cos A)\), \(A=\cos^{-1}(-0.035256)\). Using a calculator, \(A\approx92.02^{\circ}\).

Answer:

\(92.02\)