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Difficulty: MediumLaw of Sines and Law of Cosines

An architect is designing a triangular solar panel frame with side lengths measuring 1515 feet, 2424 feet, and 2121 feet. What is the measure, in degrees, of the interior angle opposite the side measuring 2121 feet?

Answer: 60 degrees

Answer

The measure of the interior angle opposite the side measuring 21 feet is 60 degrees.
Using the Law of Cosines c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab\cos(C) with side lengths a=15a = 15, b=24b = 24, and c=21c = 21 gives 212=152+242−2(15)(24)cos⁡(C)21^2 = 15^2 + 24^2 - 2(15)(24)\cos(C). Simplifying the equation leads to 441=801−720cos⁡(C)441 = 801 - 720\cos(C), which rearranges to −360=−720cos⁡(C)-360 = -720\cos(C) or cos⁡(C)=0.5\cos(C) = 0.5. Evaluating arccos⁡(0.5)\arccos(0.5) yields an angle measure of 60∘60^\circ.

Step-by-Step Solution

1
Set up the Law of Cosines with the side lengths a=15a = 15, b=24b = 24, and target opposite side c=21c = 21.
212=152+242−2(15)(24)cos⁡(C)21^2 = 15^2 + 24^2 - 2(15)(24)\cos(C)
The Law of Cosines connects three side lengths of any triangle to the cosine of the angle opposite one of those sides.
2
Simplify the numerical values in the equation.
441=801−720cos⁡(C)441 = 801 - 720\cos(C)
Evaluate 212=44121^2 = 441, 152+242=225+576=80115^2 + 24^2 = 225 + 576 = 801, and 2(15)(24)=7202(15)(24) = 720.
3
Isolate the cosine expression.
cos⁡(C)=0.5\cos(C) = 0.5
Subtracting 801801 from both sides gives −360=−720cos⁡(C)-360 = -720\cos(C), then dividing by −720-720 yields 0.50.5.
4
Find the inverse cosine of 0.50.5.
C=60∘C = 60^\circ
In any triangle, the angle whose cosine is 0.50.5 is 60∘60^\circ.

Key Concept

Applying the Law of Cosines to solve for an unknown angle given all three side lengths of a non-right triangle.
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Topics

Law of Sines and Law of Cosines

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