A scalene triangle has the lengths 6, 11, and 12. Keyla uses the law of cosines to find the measure of the largest angle. Complete her work and find the measure of angle [tex]$Y$[/tex] to the nearest degree.

1. [tex]$12^2=11^2+6^2-2(11)(6) \cos (y)$[/tex]
2. [tex]$144=121+36-(132) \cos (Y)$[/tex]
3. [tex]$144=157-(132) \cos (y)$[/tex]
4. [tex]$-13=-(132) \cos (y)$[/tex]

_____ degrees

Using the Law of Cosines, we determined that the largest angle Y in the triangle with sides 6, 11, and 12 is approximately 84 degrees. This was calculated by isolating the cosine term and using the arccosine function. The final result, rounded to the nearest degree, is 84 degrees.
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Answered by Anonymous | 2025-07-04

Apply the Law of Cosines to relate the side lengths and the largest angle Y : 1 2 2 = 1 1 2 + 6 2 − 2 ( 11 ) ( 6 ) cos ( Y ) .
Simplify the equation to isolate the cosine term: cos ( Y ) = 132 13 ​ .
Find the angle Y by taking the inverse cosine: Y = arccos ( 132 13 ​ ) .
Round the result to the nearest degree: 84 ​ degrees.

Explanation

Problem Setup and Goal We are given a scalene triangle with side lengths 6, 11, and 12. We want to find the measure of the largest angle, which we'll call Y . Keyla has already started using the Law of Cosines, and we need to complete her work.

Review of Initial Steps Keyla's initial steps are:

1 2 2 = 1 1 2 + 6 2 − 2 ( 11 ) ( 6 ) cos ( Y )

144 = 121 + 36 − ( 132 ) cos ( Y )

144 = 157 − ( 132 ) cos ( Y )

− 13 = − ( 132 ) cos ( Y )

Isolating the Cosine Now, let's continue from step 4: − 13 = − ( 132 ) cos ( Y ) Divide both sides by -132: − 132 − 13 ​ = cos ( Y ) 132 13 ​ = cos ( Y )

Finding the Angle To find the angle Y , we need to take the inverse cosine (also known as arccos) of 132 13 ​ :
Y = arccos ( 132 13 ​ )

Calculating the Angle Using a calculator, we find the value of Y in degrees: Y ≈ 84.34 degrees

Rounding to Nearest Degree Rounding to the nearest degree, we get: Y ≈ 84 degrees

Final Answer Therefore, the measure of the largest angle Y is approximately 84 degrees.


Examples
The Law of Cosines is a fundamental concept in trigonometry that helps us solve problems involving triangles when we know the lengths of all three sides or when we know two sides and the included angle. In real life, this is incredibly useful for things like surveying land, designing structures, or even navigating using GPS. For example, if you're building a bridge and need to calculate the angles and lengths of support beams, the Law of Cosines can help you ensure the structure is stable and safe. Imagine you're a surveyor mapping a plot of land; you can measure the distances between three points and then use the Law of Cosines to determine the angles, allowing you to create an accurate map.

Answered by GinnyAnswer | 2025-07-04