Question

The value of cos 54

Answer 1

cos 54 = cos ( 90°- 36° )
cos 54 = sin 36°
Ans- Sin 36°.....

Answer 2

How to find exact value of cos 54°?

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 cos33 A - 3 cos A

⇒ 2 sin A cos A - 4 cos33 A + 3 cos A = 0

⇒ cos A (2 sin A - 4 cos22 A + 3) = 0

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

⇒ 2 sin θ - 4 (1 - sin22 A) + 3 = 0

⇒ 4 sin22 A + 2 sin A - 1 = 0, which is a quadratic in sin A

Therefore, sin θ = −2±−4(4)(−1)√2(4)−2±−4(4)(−1)2(4)
⇒ sin θ = −2±4+16√8−2±4+168
⇒ sin θ = −2±25√8−2±258
⇒ sin θ = −1±5√4−1±54
Now sin 18° is positive, as 18° lies in first quadrant.

Therefore, sin 18° = sin A = −1±5√4−1±54
Now, cos 36° = cos 2 ∙ 18°

⇒ cos 36° = 1 - 2 sin22 18°

⇒ cos 36° = 1 - 2(5√−14)2(5−14)2
⇒ cos 36° = 16−2(5+1−25√)1616−2(5+1−25)16
⇒ cos 36° = 1+45√161+4516
⇒ cos 36° = 5√+145+14
Therefore, sin 36° = 1−cos236°−−−−−−−−−√1−cos236°,[Taking sin 36° is positive, as 36° lies in first quadrant, sin 36° > 0]

⇒ sin 36° = 1−(5√+14)2−−−−−−−−−−√1−(5+14)2
⇒ sin 36° = 16−(5+1+25√)16−−−−−−−−−−√16−(5+1+25)16
⇒ sin 36° = 10−25√16−−−−−−√10−2516
⇒ sin 36° = 10−25√√410−254
Therefore, sin 36° = 10−25√√410−254
Now cos 54° = cos (90° - 36°) = sin 36° = 10−25√√410−254
Therefore, cos 54° = 10−25√√4