 # How do you find the exact value of cos 36 using the sum and difference, double angle or half-angle formulas?

We need to find the exact value of cos 36 using the sum and difference, double angle or half angle formulas

## Solution

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 cos3 A – 3 cos A

⇒ 2 sin A cos A – 4 cos3 A + 3 cos A = 0

⇒ cos A (2 sin A – 4 cos2 A + 3) = 0

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

⇒ 2 sin θ – 4 (1 – sin2 A) + 3 = 0

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

Therefore, sin θ = −2±√−4(4)(−1)/2(4)
⇒ sin θ = −2±√4+16/8
⇒ sin θ = −2±2√5/8
⇒ sin θ = −1±√5/4
Now sin 18° is positive, as 18° lies in first quadrant.

Therefore, sin 18° = sin A = −1±√5/4
Now, cos 36° = cos 2 ∙ 18°

⇒ cos 36° = 1 – 2 sin2 18°

⇒ cos 36° = 1 – 2(√5−1/4)2
⇒ cos 36° = 16−2(5+1−2√5)/16
⇒ cos 36° = 1+4√5/16
⇒ cos 36° = √5+1/4

Answer
Therefore, cos 36° = √5+1/4

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