Question

What is the value of sin18​

Answer 1

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Let A = 18°                          

Therefore, 5A = 90° 

⇒ 2A + 3A = 90˚

⇒ 2θ = 90˚ - 3A

Taking sine on both sides, we get 

sin 2A = sin (90˚ - 3A) = cos 3A 

⇒ 2 sin A cos A = 4 cos^3 A - 3 cos A

⇒ 2 sin A cos A - 4 cos^3A + 3 cos A = 0 

⇒ cos A (2 sin A - 4 cos^2 A + 3) = 0 

Dividing both sides by cos A = cos 18˚ ≠ 0, we get

⇒ 2 sin θ - 4 (1 - sin^2 A) + 3 = 0

⇒ 4 sin^2 A + 2 sin A - 1 = 0, which is a quadratic in sin A

Therefore, sin θ = −2±−4(4)(−1)√2(4)−2±−4(4)(−1)2(4)

⇒ sin θ = −2±4+16√8−2±4+168

⇒ sin θ = −2±25√8−2±258

⇒ sin θ = −1±5√4−1±54

Now sin 18° is positive, as 18° lies in first quadrant.

Therefore, sin 18° = sin A = −1±5√4−1±54

Answer 2

$\bold {\huge {Your ~answer :-}}$

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$sin18 \: = \frac{ \sqrt{5} - 1 }{4} = cos72$

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