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Exact Value of sin 72°

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


How to find the exact value of sin 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±258

⇒ sin A = 1±54

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

Therefore, sin 18° = sin A = 514

And cos 18° = √(1 - sin2 18°), [Taking positive value, cos 18° > 0]

⇒ cos 18° = 1(514)2

⇒ cos 18° = 16(5+125)16

⇒ cos 18° =  10+2516

Therefore, cos 18° = 10+254

Now sin 72° = sin (90° - 18°) = cos 18° = 10+254

 Submultiple Angles






11 and 12 Grade Math

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