Question

solve for x.
x =
law of cosines: c² = a² + b² - 2ab·cos(c)
round your answer to the nearest hundredth

Explanation:

Step1: Identify sides for Law of Cosines

In the triangle, the sides adjacent to angle \( x \) are \( a = 50 \), \( b = 90 \), and the side opposite angle \( x \) is \( c = 55 \). The Law of Cosines formula is \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( C = x \). So we rearrange to solve for \( \cos(x) \):
\( \cos(x) = \frac{a^2 + b^2 - c^2}{2ab} \)

Step2: Substitute values into the formula

Substitute \( a = 50 \), \( b = 90 \), \( c = 55 \):
\( \cos(x) = \frac{50^2 + 90^2 - 55^2}{2 \times 50 \times 90} \)
Calculate numerator: \( 2500 + 8100 - 3025 = 7575 \)
Calculate denominator: \( 2 \times 50 \times 90 = 9000 \)
So \( \cos(x) = \frac{7575}{9000} = 0.841666\ldots \)

Step3: Find \( x \) using arccosine

Take the arccosine (inverse cosine) of \( 0.841666\ldots \):
\( x = \arccos(0.841666\ldots) \)
Using a calculator, \( x \approx 32.68^\circ \) (rounded to the nearest hundredth)

Answer:

\( 32.68 \)