The law of cosines is used to find the measure of $\angle 2$.
$\begin{array}{l}<>16^2=18^2+19^2-2(18)(19) \cos (Z) \\
256=324+361-(684) \cos (Z) \\
256=685-(684) \cos (Z)<>\end{array}$

To the nearest whole degree, what is the measure of $\angle Z$?
A. $41^{\circ}$
B. $47^{\circ}$
C. $51^{\circ}$
D. $57^{\circ}

Using the law of cosines, we isolate cos ( Z ) and find it to be approximately 0.6272. Taking the inverse cosine gives us an angle of approximately 51.1 degrees, which rounds to 51 degrees. Therefore, the measure of angle Z is 5 1 ∘ .
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Answered by Anonymous | 2025-07-04

Isolate the term with cos ( Z ) in the given equation: 684 cos ( Z ) = 429 .
Solve for cos ( Z ) : cos ( Z ) = 684 429 ​ ≈ 0.6272 .
Find the angle Z by taking the inverse cosine: Z = arccos ( 0.6272 ) ≈ 51. 1 ∘ .
Round the result to the nearest whole degree: 5 1 ∘ ​ .

Explanation

Analyze the problem and equation. We are given the equation derived from the law of cosines: 256 = 685 − 684 cos ( Z ) Our goal is to find the measure of angle Z to the nearest whole degree.

Isolate the cosine term. First, we isolate the term with cos ( Z ) :
684 cos ( Z ) = 685 − 256 684 cos ( Z ) = 429

Solve for cosine of Z. Next, we solve for cos ( Z ) :
cos ( Z ) = 684 429 ​ cos ( Z ) = 0.6272

Find angle Z using arccos. Now, we find the angle Z by taking the inverse cosine (arccos) of the value we just found: Z = arccos ( 0.6272 ) Z ≈ 51.1 degrees

Round to the nearest degree. Finally, we round the result to the nearest whole degree: Z ≈ 5 1 ∘


Examples
The law of cosines is a fundamental concept in trigonometry that helps us solve for unknown angles and sides in triangles when we have certain information. For example, imagine you are designing a bridge, and you need to calculate the angles and lengths of the supporting triangular structures. By knowing the lengths of the sides of a triangle, you can use the law of cosines to find the angles needed to ensure the bridge is stable and safe. This principle is also used in navigation, surveying, and various engineering applications to accurately determine distances and angles.

Answered by GinnyAnswer | 2025-07-04