Trigonometry! #52

Geometry Level 3

If the angles A A , B B and C C of a Δ A B C \Delta ABC are in A.P. and b : c = 3 : 2 b:c=\sqrt{3}:\sqrt{2} , then find A A . Enter your answer in degrees, without the degree sign.

This problem is part of the set Trigonometry .


The answer is 75.

Omkar Kulkarni by Omkar Kulkarni , 22, India

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1 solution

Omkar Kulkarni
Feb 14, 2015

As the angles A A , B B , C C are in AP, B = π 3 B=\frac{\pi}{3}

∴ b c = 3 2 sin ⁡ B sin ⁡ C = 3 2 sin ⁡ π 3 sin ⁡ C = 3 2 sin ⁡ C = 1 2 ⇒ C = π 4 \large{\therefore \frac{b}{c}=\frac{\sqrt{3}}{\sqrt{2}} \\ \frac{\sin B}{\sin C}=\frac{\sqrt{3}}{\sqrt{2}} \\ \frac{\sin \frac{\pi}{3}}{\sin C}=\frac{\sqrt{3}}{\sqrt{2}} \\ \sin C = \frac{1}{\sqrt{2}} \Rightarrow C=\frac{\pi}{4}}

∴ A = 5 π 12 = 7 5 ∘ \large{\therefore A = \frac {5\pi}{12} = 75^{\circ}}

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