Question

find the measure of angle c.
b
c = 90
a
c = ?°
b = 50
law of cosines: c² = a² + b² - 2ab·cos(c)
round your answer to the nearest hundredth.
submit

Explanation:

Step1: Substitute values into Law of Cosines

We know \(a = 55\), \(b = 50\), \(c = 90\) and the Law of Cosines formula \(c^{2}=a^{2}+b^{2}-2ab\cdot\cos(C)\). Substitute the values:

$$ 90^{2}=55^{2}+50^{2}-2\times55\times50\times\cos(C) $$

Calculate each term: \(90^{2}=8100\), \(55^{2} = 3025\), \(50^{2}=2500\). So:

$$ 8100=3025 + 2500-5500\times\cos(C) $$

Step2: Simplify the equation

First, add \(3025\) and \(2500\): \(3025 + 2500=5525\). Then the equation becomes:

$$ 8100=5525-5500\times\cos(C) $$

Subtract \(5525\) from both sides: \(8100 - 5525=- 5500\times\cos(C)\), so \(2575=-5500\times\cos(C)\)

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

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

Step4: Find angle \(C\)

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

Answer:

\(117.87\)