Question

find the value of 1 ) cos 18​

Answer 1

Let, A = 18°                        

Therefore, 5A = 90°

⇒ 2A + 3A = 90˚

⇒ 2A = 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 $cos^{3}A$ + 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 A - 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 A = −2±−4(4)(−1)√2(4)

⇒ sin A = −2±4+16√8

⇒ sin A = −2±25√8

⇒ sin A = −1±5√4

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

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

Now cos 18° = $\sqrt{1-sin^{2}18 }$, [Taking positive value, cos 18° > 0]

⇒ cos 18° = $\sqrt{1-((\sqrt{5}-1)/4)2 }$

⇒ cos 18° = $\sqrt{(16-(5+1-2\sqrt{5}))/16 }$

⇒ cos 18° = $\sqrt{(10+2\sqrt{5}) } /4$

Therefore, cos 18° = $\sqrt{(10+2\sqrt{5}) } /4$

TAKE CARE.

HOPE THIS WILL HELP YOU.