Find the value of sin 30° geometrically.

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Solution

Consider an equilateral ∆ABC with each side equal to 2a.
Then, each angle of ∆ABC = 60^o
From A, draw AD ⊥ BC.
Figure
Then, it is clear that BD = DC = a
Also, ∠ADB = 90^o
Now, in ∆ADB, we have:
∠ADB + ∠DBA + ∠BAD = 180^o (Angle sum property)
⇒ 90^o + 60^o + ∠BAD = 180^o
⇒ ∠BAD = 180^o − 150^o
⇒ ∠BAD = 30^o
∴ sin 30^o = $\frac{B D}{A B} = \frac{a}{2 a} = \frac{1}{2}$

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