Exact Value of cos 22½°

How to find the exact value of cos 22½° using the value of cos 45°?

Solution:

22½° lies in the first quadrant.

Therefore, sin 22½° is positive.

For all values of the angle A we know that, cos A = 2 cos\(^{2}\) \(\frac{A}{2}\)  - 1

⇒ 1 + cos A = 2 cos\(^{2}\) \(\frac{A}{2}\)

⇒ 2 cos\(^{2}\) \(\frac{A}{2}\) = 1 + cos A

⇒ 2 cos\(^{2}\) 22½˚ = 1 + cos 45°

⇒ cos\(^{2}\) 22½˚ = \(\frac{1 + cos 45°}{2}\)

⇒ sin\(^{2}\) 22½˚ = \(\frac{1 + \frac{1}{\sqrt{2}}}{2}\), [Since we know cos 45° = \(\frac{1}{√2}\)]

⇒ cos 22½˚ = \(\sqrt{\frac{1}{2}(1 +  \frac{1}{\sqrt{2}})}\), [Since, cos 22½˚ > 0]

⇒ cos 22½˚ = \(\sqrt{\frac{\sqrt{2} + 1}{2\sqrt{2}}}\)

⇒ cos 22½˚ = \(\frac{1}{2}\sqrt{2 + \sqrt{2}}\)

Therefore, cos 22½˚ = \(\frac{1}{2}\sqrt{2 + \sqrt{2}}\)

● Submultiple Angles

• Trigonometric Ratios of Angle A2A2
• Trigonometric Ratios of Angle A3A3
• Trigonometric Ratios of Angle A2A2 in Terms of cos A
• tan A2A2 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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