Question

solve for x.
x = ?
law of cosines: c² = a² + b² - 2ab·cos(c)
round your answer to the nearest hundredth.
(image of a triangle with sides 8, 16, 17 and angle x° at the vertex between sides 16 and 17)

Explanation:

Step1: Identify sides for Law of Cosines

Let \( a = 16 \), \( b = 8 \), \( c = 17 \). The angle opposite side \( c \) is \( x \), so use Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos(x) \).
Substitute values: \( 17^2 = 16^2 + 8^2 - 2(16)(8)\cos(x) \).

Step2: Simplify the equation

Calculate squares: \( 289 = 256 + 64 - 256\cos(x) \).
Simplify right side: \( 289 = 320 - 256\cos(x) \).
Rearrange: \( 256\cos(x) = 320 - 289 \).
Calculate: \( 256\cos(x) = 31 \).

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

Divide: \( \cos(x) = \frac{31}{256} \approx 0.1211 \).

Step4: Find \( x \) using arccos

\( x = \arccos(0.1211) \approx 83.09^\circ \).

Answer:

\( 83.09 \)