Exact Value of tan 27°

We will learn to find the exact value of tan 27 degrees using the formula of submultiple angles.

How to find the exact value of tan 27°?

Solution: 

We have, (sin 27° + cos 27°)$^{2}$ = sin$^{2}$ 27° + cos$^{2}$ 27° + 2 sin 27° cos 27°

⇒ (sin 27° + cos 27°)$^{2}$ = 1+ sin 2 ∙ 27°

⇒ (sin 27° + cos 27°)$^{2}$ = 1 + sin 54° 

⇒ (sin 27° + cos 27°)$^{2}$ = 1 + sin (90° - 36°)

⇒ (sin 27° + cos 27°)$^{2}$ = 1 + cos 36° 

⇒ (sin 27° + cos 27°)$^{2}$ = 1+ $\frac{√5 + 1}{4}$

⇒ (sin 27° + cos 27°)$^{2}$ = $\frac{1}{4}$ ( 5 + √ 5)

Therefore,  sin 27° + cos 27° = $\frac{1}{2}\sqrt{5 + \sqrt{5}}$ …………….….(i)

[Since, sin 27° > 0 and cos 27° > 0)

Similarly, we have, 

(sin 27° - cos 27°)$^{2}$ = 1 - cos 36°

⇒ (sin 27° - cos 27°)$^{2}$ = 1 - $\frac{√5 +1}{4}$

⇒ (sin 27° - cos 27°)$^{2}$ = $\frac{1}{4}$ (3 - √5  )
Therefore, sin 27° - cos 27° = ± $\frac{1}{2}\sqrt{3 - \sqrt{5}}$ …………….….(ii)
Now, sin 27° - cos 27° = √2 ($\frac{1}{√2}$ sin 27˚ - $\frac{1}{√2}$ cos 27°)

                               =√2 (cos 45° sin 27° - sin 45° cos 27°)

                               = √2 sin (27° - 45°)

                               = -√2 sin 18° < 0

Therefore, from (ii) we get,

sin 27° - cos 27° = -$\frac{1}{2}\sqrt{3 - \sqrt{5}}$ …………….….(iii)

Now, adding (i) and (iii) we get,

2 sin 27° = $\frac{1}{2}\sqrt{5 + \sqrt{5}}$ - $\frac{1}{2}\sqrt{3 - \sqrt{5}}$

⇒ sin 27° = $\frac{1}{4}(\sqrt{5 + \sqrt{5}} - \sqrt{3 - \sqrt{5}})$

Therefore, sin 27° = $\frac{1}{4}(\sqrt{5 + \sqrt{5}} - \sqrt{3 - \sqrt{5}})$…………….….(iv)

Again, subtracting (iii) and (i) we get,

2 cos 27° = $\frac{1}{2}\sqrt{5 + \sqrt{5}}$ + $\frac{1}{2}\sqrt{3 - \sqrt{5}}$

⇒ cos 27° = $\frac{1}{4}(\sqrt{5 + \sqrt{5}} + \sqrt{3 - \sqrt{5}})$

Therefore, cos 27° = $\frac{1}{4}(\sqrt{5 + \sqrt{5}} + \sqrt{3 - \sqrt{5}})$…………….….(v)

Now dividing (iv) by (v) we get,

tan 27° = $\frac{\sqrt{5 + \sqrt{5}} - \sqrt{3 - \sqrt{5}}}{\sqrt{5 + \sqrt{5}} + \sqrt{3 - \sqrt{5}}}$

● 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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