Question

solve for x.
law of cosines: ( c^2 = a^2 + b^2 - 2ab cdot cos(c) )
round your answer to the nearest hundredth.
(image of a triangle with sides 55, 50, 90 and angle ( x^circ ) between sides 55 and 50)

Explanation:

Step1: Identify sides for Law of Cosines

In the triangle, the side opposite angle \( x \) is \( 90 \), and the other two sides are \( a = 55 \), \( b = 50 \), \( c = 90 \). The Law of Cosines is \( c^{2}=a^{2}+b^{2}-2ab\cos(C) \), where \( C = x \). So we substitute: \( 90^{2}=55^{2}+50^{2}-2\times55\times50\times\cos(x) \).

Step2: Simplify the equation

Calculate \( 90^{2}=8100 \), \( 55^{2}=3025 \), \( 50^{2}=2500 \). Then the equation becomes \( 8100 = 3025 + 2500 - 5500\cos(x) \). Combine like terms: \( 8100 = 5525 - 5500\cos(x) \).

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

Subtract \( 5525 \) from both sides: \( 8100 - 5525=- 5500\cos(x) \), so \( 2575=-5500\cos(x) \). Then \( \cos(x)=\frac{2575}{- 5500}\approx - 0.4682 \).

Step4: Find \( x \) using arccos

Take the arccosine: \( x=\arccos(-0.4682) \). Using a calculator, \( x\approx117.87^{\circ} \) (rounded to the nearest hundredth).

Answer:

\( 117.87 \)