Question

find the measure of \\( \angle a \\). round to the nearest tenth of a degree.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\). Let \(a = 180\), \(b = 150\), \(c = 170\). Then \(180^{2}=150^{2}+170^{2}-2\times150\times170\times\cos A\).

Step2: Simplify the equation

First, calculate the squares: \(32400=22500 + 28900-51000\cos A\). Then \(32400=51400-51000\cos A\).

Step3: Solve for \(\cos A\)

Rearrange the equation: \(51000\cos A=51400 - 32400\). So \(51000\cos A=19000\), and \(\cos A=\frac{19000}{51000}\approx0.3725\).

Step4: Find the angle \(A\)

Take the inverse cosine: \(A=\cos^{-1}(0.3725)\approx68.1^{\circ}\).

Answer:

\(68.1^{\circ}\)