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Exact Value of tan 54°

We will learn to find the exact value of tan 54 degrees using the formula of multiple angles. 


How to find exact value of tan 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 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+168

⇒ sin θ = 2±258

⇒ sin θ = 1±54

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

Therefore, sin 18° = sin A = 1±54

Now, cos 36° = cos 2 ∙ 18°

⇒ cos 36° = 1 - 2 sin2 18°

⇒ cos 36° = 1 - 2(514)2

⇒ cos 36° = 162(5+125)16

⇒ cos 36° = 1+4516

⇒ cos 36° = 5+14

Therefore, sin 36° = 1cos236°,[Taking sin 36° is positive, as 36° lies in first quadrant, sin 36° > 0]

⇒ sin 36° = 1(5+14)2

⇒ sin 36° = 16(5+1+25)16

⇒ sin 36° = 102516

⇒ sin 36° = 10254

Therefore, sin 36° = 10254

Now sin 54° = sin (90° - 36°) = cos 36° = 5+14

Similarly, cos 54° = cos (90° - 36°) = sin 36° = 10254

Therefore, tan 54° = sin54°cos54°

⇒ tan 54° =   5+1410254

⇒ tan 54° = 5+11025

Therefore, tan 54° = 5+11025.

 Submultiple Angles






11 and 12 Grade Math

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