Question

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

Explanation:

Step1: Substitute values into Law of Cosines

We know \( c = 55 \), \( a = 50 \), \( b = 90 \). Substitute into \( c^{2}=a^{2}+b^{2}-2ab\cdot\cos(C) \):
\( 55^{2}=50^{2}+90^{2}-2\times50\times90\times\cos(C) \)
Calculate each term: \( 55^{2} = 3025 \), \( 50^{2}=2500 \), \( 90^{2}=8100 \)
So \( 3025 = 2500 + 8100 - 9000\cos(C) \)

Step2: Simplify the equation

Simplify the right - hand side: \( 2500 + 8100=10600 \)
The equation becomes \( 3025=10600 - 9000\cos(C) \)
Rearrange to solve for \( \cos(C) \):
\( 9000\cos(C)=10600 - 3025 \)
\( 9000\cos(C)=7575 \)
\( \cos(C)=\frac{7575}{9000}=\frac{101}{120}\approx0.8417 \)

Step3: Find angle C

Take the inverse cosine: \( C=\cos^{-1}(0.8417) \)
Using a calculator, \( C\approx32.6^{\circ} \) (rounded to one decimal place)

Answer:

\( \approx32.6^{\circ} \) (If we need to round to the nearest whole number, it is \( 33^{\circ} \))