Question

solve the triangle.
a = 62, b = 23, c = 51
a ≈ 107.7°
(do not round until the final answer. then round to the nearest degree as needed.)

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines formula is \( \cos A=\frac{b^{2}+c^{2}-a^{2}}{2bc}\).
Substitute \(a = 62\), \(b = 23\), \(c = 51\) into the formula:
\(\cos A=\frac{23^{2}+51^{2}-62^{2}}{2\times23\times51}\)
First, calculate the squares: \(23^{2}=529\), \(51^{2}=2601\), \(62^{2}=3844\)
Then, calculate the numerator: \(529 + 2601-3844=529+2601 - 3844=-714\)
The denominator is \(2\times23\times51 = 2346\)
So, \(\cos A=\frac{-714}{2346}\approx - 0.3043\)

Step2: Find angle \(A\)

Since \(A=\cos^{-1}(\cos A)\), then \(A=\cos^{-1}(-0.3043)\)
Using a calculator, \(A\approx107.7^{\circ}\)

Answer:

\(A\approx108^{\circ}\) (rounded to the nearest degree)