Exact Value of tan 72°

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

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 cos$^{3}$ A - 3 cos A

⇒ 2 sin A cos A - 4 cos$^{3}$ A + 3 cos A = 0

⇒ cos A (2 sin A - 4 cos$^{2}$ A + 3) = 0 

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

⇒ 2 sin A - 4 (1 - sin$^{2}$ A) + 3 = 0

⇒ 4 sin$^{2}$ A + 2 sin A - 1 = 0, which is a quadratic in sin A

Therefore, sin A = $\frac{-2 \pm \sqrt{- 4 (4)(-1)}}{2(4)}$

⇒ sin A = $\frac{-2 \pm \sqrt{4 + 16}}{8}$

⇒ sin A = $\frac{-2 \pm 2 \sqrt{5}}{8}$

⇒ sin A = $\frac{-1 \pm \sqrt{5}}{4}$

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

Therefore, sin 18° = sin A = $\frac{√5 - 1}{4}$

Now, cos 72° = cos (90° - 18°) = sin 18° = $\frac{√5 - 1}{4}$

And cos 18° = √(1 - sin$^{2}$ 18°), [Taking positive value, cos 18° > 0]

⇒ cos 18° = $\sqrt{1 - (\frac{\sqrt{5} - 1}{4})^{2}}$

⇒ cos 18° = $\sqrt{\frac{16 - (5 + 1 - 2\sqrt{5})}{16}}$

⇒ cos 18° =  $\sqrt{\frac{10 + 2\sqrt{5}}{16}}$

Thus, sin 72° = sin (90° - 18°) = cos 18° = $\frac{\sqrt{10 + 2\sqrt{5}}}{4}$

Now, tan 72° = $\frac{sin 72°}{cos 72°}$ = $\frac{\frac{\sqrt{10 + 2\sqrt{5}}}{4}}{\frac{√5 - 1}{4}}$ = $\frac{\sqrt{10 + 2√5}}{√5 - 1}$

Therefore, tan 72° =$\frac{\sqrt{10 + 2√5}}{√5 - 1}$

● Submultiple Angles

• Trigonometric Ratios of Angle $\frac{A}{2}$
• Trigonometric Ratios of Angle $\frac{A}{3}$
• Trigonometric Ratios of Angle $\frac{A}{2}$ in Terms of cos A
• tan $\frac{A}{2}$ in Terms of tan A
• Exact value of sin 7½°
• Exact value of cos 7½°
• Exact value of tan 7½°
• Exact Value of cot 7½°
• Exact Value of tan 11¼°
  
• Exact Value of sin 15°
• Exact Value of cos 15°
• Exact Value of tan 15°
  
• Exact Value of sin 18°
• Exact Value of cos 18°
• Exact Value of sin 22½°
• Exact Value of cos 22½°
• Exact Value of tan 22½°
• Exact Value of sin 27°
• Exact Value of cos 27°
• Exact Value of tan 27°
  
• Exact Value of sin 36°
• Exact Value of cos 36°
• Exact Value of sin 54°
• Exact Value of cos 54°
• Exact Value of tan 54°
• Exact Value of sin 72°
• Exact Value of cos 72°
• Exact Value of tan 72°
• Exact Value of tan 142½°
  
• Submultiple Angle Formulae
  
• Problems on Submultiple Angles

11 and 12 Grade Math

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