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Exact Value of cos 72°

We will learn to find the exact value of cos 72 degrees using the formula of submultiple angles.


How to find the exact value of cos 72°?

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 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 A - 4 (1 - sin2 A) + 3 = 0

⇒ 4 sin2 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+168

⇒ sin A = −2±2√58

⇒ sin A = −1±√54

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

Therefore, sin 18° = sin A = √5−14

Now, cos 72° = cos (90° - 18°) = sin 18° = √5−14



● Submultiple Angles





11 and 12 Grade Math

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