Question

the law of cosines is used to find the measure of ∠z. to the nearest whole degree, what is the measure of ∠z? 16² = 18² + 19² - 2(18)(19)cos(z) 256 = 324 + 361 - (684)cos(z) 256 = 685 - (684)cos(z) - 429 = -(684)cos(z)

Explanation:

Step1: Law of Cosines formula

$XY^2 = XZ^2 + YZ^2 - 2(XZ)(YZ)\cos(Z)$

Step2: Substitute values

$16^2 = 18^2 + 19^2 - 2(18)(19)\cos(Z)$

Step3: Calculate squares

$256 = 324 + 361 - 684\cos(Z)$

Step4: Simplify right side

$256 = 685 - 684\cos(Z)$

Step5: Solve for cos(Z)

$-429 = -684\cos(Z) \Rightarrow \cos(Z) = 429/684 \approx 0.627$

Step6: Find angle Z

$Z = \arccos(0.627) \approx 51^\circ$

Answer:

C. 51°