Question

solve for x.
law of cosines: ( c^2 = a^2 + b^2 - 2ab cdot cos(c) )
round your answer to the nearest hundredth
( x = \boxed{?}^circ )

Explanation:

Step1: Identify sides for Law of Cosines

In the triangle, the sides adjacent to angle \( x \) are \( a = 16 \), \( b = 8 \), and the side opposite angle \( x \) is \( c = 17 \). The Law of Cosines formula is \( c^2 = a^2 + b^2 - 2ab\cos(C) \), where \( C = x \). So we substitute the values: \( 17^2 = 16^2 + 8^2 - 2(16)(8)\cos(x) \).

Step2: Simplify the equation

Calculate the squares: \( 289 = 256 + 64 - 256\cos(x) \). Then simplify the right - hand side: \( 289 = 320 - 256\cos(x) \).

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

Rearrange the equation to isolate \( \cos(x) \): \( 256\cos(x)=320 - 289 \). So \( 256\cos(x) = 31 \). Then \( \cos(x)=\frac{31}{256}\approx0.1211 \).

Step4: Find \( x \) using arccos

Take the arccosine of both sides: \( x=\arccos(0.1211) \). Using a calculator, \( x\approx83.13^{\circ} \) (rounded to the nearest hundredth).

Answer:

\( 83.13 \)