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Difficulty: MediumSine and Cosine Rules

In ΔKLM\Delta KLM, side k=5 cmk = 5\text{ cm}, side l=53 cml = 5\sqrt{3}\text{ cm}, and ∠K=30∘\angle K = 30^\circ. If ∠L\angle L is an obtuse angle, what is the measure of ∠L\angle L?

  1. A
    60∘60^\circ
  2. 120∘120^\circAnswer
  3. C
    135∘135^\circ
  4. D
    150∘150^\circ

Answer

The measure of angle LL is 120∘120^\circ.
Applying the Sine Rule ksin⁡K=lsin⁡L\frac{k}{\sin K} = \frac{l}{\sin L} gives sin⁡L=53⋅sin⁡30∘5=32\sin L = \frac{5\sqrt{3} \cdot \sin 30^\circ}{5} = \frac{\sqrt{3}}{2}. The inverse sine gives an acute angle of 60∘60^\circ. Because the problem specifies that angle LL is obtuse, we find its supplementary angle in the second quadrant: 180∘−60∘=120∘180^\circ - 60^\circ = 120^\circ.

Step-by-Step Solution

1
Apply the Sine Rule relating sides k,lk, l and angles K,LK, L.
ksin⁡K=lsin⁡L\frac{k}{\sin K} = \frac{l}{\sin L}
The Sine Rule connects the ratio of side lengths to the sines of their opposite angles.
2
Substitute the given values into the Sine Rule equation.
5sin⁡30∘=53sin⁡L  ⟹  50.5=53sin⁡L  ⟹  10=53sin⁡L\frac{5}{\sin 30^\circ} = \frac{5\sqrt{3}}{\sin L} \implies \frac{5}{0.5} = \frac{5\sqrt{3}}{\sin L} \implies 10 = \frac{5\sqrt{3}}{\sin L}
Known values are k=5 cmk = 5\text{ cm}, l=53 cml = 5\sqrt{3}\text{ cm}, and sin⁡30∘=12\sin 30^\circ = \frac{1}{2}.
3
Solve for sin⁡L\sin L.
sin⁡L=5310=32\sin L = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}
Isolating sin⁡L\sin L yields the principal ratio.
4
Determine the obtuse angle solution for LL.
L=180∘−60∘=120∘L = 180^\circ - 60^\circ = 120^\circ
Since sin⁡L=32\sin L = \frac{\sqrt{3}}{2}, the acute solution is 60∘60^\circ, so the supplementary obtuse solution is 180∘−60∘=120∘180^\circ - 60^\circ = 120^\circ.

Key Concept

Ambiguous Case of the Sine Rule
Estimated Time:1m 15s
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Topics

Sine and Cosine Rules

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