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Question

Difficulty: MediumSine and Cosine Rules

In triangle LMNLMN, the side lengths are given as l=7 cml = 7\text{ cm}, m=8 cmm = 8\text{ cm}, and n=13 cmn = 13\text{ cm}. What is the measure of angle NN in degrees?

Answer: 120 degrees

Answer

The measure of angle NN is 120∘120^\circ.
Using the Cosine Rule for angle NN, cos⁡N=l2+m2−n22lm=72+82−1322(7)(8)=−56112=−0.5\cos N = \frac{l^2 + m^2 - n^2}{2lm} = \frac{7^2 + 8^2 - 13^2}{2(7)(8)} = \frac{-56}{112} = -0.5. The angle whose cosine is −0.5-0.5 within the interior angles of a triangle (0∘<N<180∘0^\circ < N < 180^\circ) is 120∘120^\circ.

Step-by-Step Solution

1
Set up the Cosine Rule formula for angle NN
\cos N = \frac{l^2 + m^2 - n^2}{2lm}
The Cosine Rule relates the three sides of any triangle to the cosine of one of its interior angles.
2
Substitute the known side lengths into the formula
\cos N = \frac{7^2 + 8^2 - 13^2}{2 \times 7 \times 8} = \frac{49 + 64 - 169}{112}
Side n=13 cmn = 13\text{ cm} is opposite to angle NN and must be subtracted in the numerator.
3
Simplify the fraction
\cos N = \frac{-56}{112} = -0.5
Evaluating the numerical expression gives a negative value, indicating that angle NN is obtuse.
4
Calculate the principal inverse cosine angle for the triangle
N=120∘N = 120^\circ
Since cos⁡60∘=0.5\cos 60^\circ = 0.5, cos⁡(180∘−60∘)=−0.5\cos(180^\circ - 60^\circ) = -0.5, giving N=120∘N = 120^\circ.

Key Concept

Applying the Cosine Rule to find an obtuse angle given three side lengths (SSS)
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Topics

Sine and Cosine Rules

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