Question

find the measure of angle c.
law of cosines: $c^{2}=a^{2}+b^{2}-2abcdot\cos c$
round your final answer to the nearest tenth.

Explanation:

Step1: Substitute values into the formula

Given \(a = 50\), \(b = 90\), \(c = 55\), substitute into \(c^{2}=a^{2}+b^{2}-2ab\cos C\).
So, \(55^{2}=50^{2}+90^{2}-2\times50\times90\times\cos C\).
Calculate \(55^{2}=3025\), \(50^{2}=2500\), \(90^{2}=8100\), \(2\times50\times90 = 9000\).
The equation becomes \(3025=2500 + 8100-9000\cos C\).

Step2: Simplify the equation

First, simplify the right - hand side: \(2500 + 8100=10600\).
The equation is \(3025=10600-9000\cos C\).
Rearrange to get \(9000\cos C=10600 - 3025\).
\(10600 - 3025 = 7575\), so \(9000\cos C=7575\).
Then \(\cos C=\frac{7575}{9000}\).
Calculate \(\frac{7575}{9000}=0.84166\cdots\).

Step3: Find the angle

Since \(C=\cos^{-1}(0.84166\cdots)\).
Using a calculator, \(C\approx32.7^{\circ}\).

Answer:

\(32.7^{\circ}\)