Question

find the measure of angle c.
b
c = 90
a = 55
c = ?°
b = 50
a
law of cosines: $c^2 = a^2 + b^2 - 2ab \cdot \cos(c)$
round your answer to the nearest hundredth.

Explanation:

Step1: Substitute values into Law of Cosines

Given \( a = 55 \), \( b = 50 \), \( c = 90 \), and the Law of Cosines \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \). Substitute the values:
\( 90^{2}=55^{2}+50^{2}-2(55)(50)\cos(C) \)
Calculate each term: \( 8100 = 3025 + 2500 - 5500\cos(C) \)

Step2: Simplify the equation

Simplify the right - hand side: \( 8100=5525 - 5500\cos(C) \)
Subtract 5525 from both sides: \( 8100 - 5525=-5500\cos(C) \)
\( 2575=-5500\cos(C) \)

Step3: Solve for \( \cos(C) \)

Divide both sides by - 5500: \( \cos(C)=\frac{2575}{-5500}\approx - 0.46818 \)

Step4: Find the angle \( C \)

Take the inverse cosine: \( C=\cos^{-1}(-0.46818) \)
Using a calculator, \( C\approx117.87^{\circ} \)

Answer:

\( 117.87^{\circ} \)