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Difficulty: MediumSine and Cosine Rules

In ΔLMN\Delta LMN, the length of side l=10 cml = 10\text{ cm}, side m=103 cmm = 10\sqrt{3}\text{ cm}, and angle ∠L=30∘\angle L = 30^\circ. Given that angle ∠M\angle M is an obtuse angle, what is the measure of angle ∠M\angle M?

  1. 120∘120^\circAnswer
  2. B
    60∘60^\circ
  3. C
    150∘150^\circ
  4. D
    45∘45^\circ

Answer

The measure of angle ∠M\angle M is 120∘120^\circ.
Using the Sine Rule, we find sin⁡M=103⋅sin⁡30∘10=32\sin M = \frac{10\sqrt{3} \cdot \sin 30^\circ}{10} = \frac{\sqrt{3}}{2}. The inverse sine gives a reference angle of 60∘60^\circ. Since the problem explicitly states that angle ∠M\angle M is obtuse, we select 180∘−60∘=120∘180^\circ - 60^\circ = 120^\circ.

Step-by-Step Solution

1
Apply the Sine Rule relating sides l,ml, m and their opposite angles L,ML, M.
lsin⁡L=msin⁡M\frac{l}{\sin L} = \frac{m}{\sin M}
The Sine Rule allows us to find an unknown angle given two sides and one non-included opposite angle.
2
Substitute the given values into the formula and solve for sin⁡M\sin M.
\sin M = \frac{10\sqrt{3} \cdot \sin 30^\circ}{10} = \sqrt{3} \cdot 0.5 = \frac{\sqrt{3}}{2}
Since sin⁡30∘=12\sin 30^\circ = \frac{1}{2}, simplifying the fraction gives 32\frac{\sqrt{3}}{2}.
3
Determine the obtuse angle solution for sin⁡M=32\sin M = \frac{\sqrt{3}}{2}.
\angle M = 180^\circ - 60^\circ = 120^\circ
The principal value is 60∘60^\circ, but because ∠M\angle M is specified to be obtuse (90∘<∠M<180∘90^\circ < \angle M < 180^\circ), we take the supplementary angle in the second quadrant.

Key Concept

Sine Rule and the Ambiguous Case (SSA Condition)
Estimated Time:1m 30s
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Topics

Sine and Cosine Rules

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