Question

what is the value of sin18'? find it without using calculator means process wise. Plz any 1 gv me the complete soln.

Answer 1

Let A=18°                          

∴ 5A=90° 

⇒ 2A+3A = 90˚

⇒ 2θ=90˚-3A

Taking sine on both sides

sin 2A = sin (90˚ - 3A) = cos 3A 

⇒ 2 sin A cos A = 4cos³ A - 3 cos A

⇒ 2 sin A cos A - 4cos³A + 3 cos A = 0 

⇒ cos A (2 sin A - 4 cos² A + 3) = 0 

dividing both sides by cos A = cos 18˚ ≠ 0

⇒ 2 sin θ - 4 (1 - sin² A) + 3 = 0

⇒ 4 sin² A + 2 sin A - 1 = 0, which is a quadratic in sin A

∴ sin θ = -2 ± $\sqrt{4+16} /8$

   sin θ = (-2 ± 2$\sqrt{5}$) / 8

    sin θ = (-1 ± $\sqrt{5}) /4$

∴ sin 18° is positive as 18° lies in first quadrant.

∴ sin 18° = sin A =( -1 ± $\sqrt{5}) /4$