Trigonometric Ratios of Angle A/2 in Terms of cos A

We will learn about the trigonometric ratios of angle A/2 in terms of cos A.

How to express sin A/2, cos A/2 and tan A/2 in terms of cos A?

(i) For all values of the angle A we know that, cos A = 2 cos^2 A/2 - 1

⇒ 2 cos^2 A/2 = 1 + cos A

⇒ cos^2 A/2 = (1 + cos A)/2

⇒ cos $\frac{A}{2}$ = ± $\sqrt{\frac{1 + cos A}{2}}$ 

(ii) For all values of the angle A we know that, cos A = 1 - 2 sin^2 A/2

⇒ 2 sin^2 A/2 = 1 - cos A

⇒ sin^2 A/2 = (1 - cos A)/2

⇒ sin A/2 = ± √{(1 - cos A)/2}

(iii) For all values of the angle A we know that, tan A/2 = sin A/2/cos A/2

⇒ tan A/2 = ± √{(1 - cos A)/2}/√{(1 + cos A)/2}

⇒ tan $\frac{A}{2}$ = ± $\sqrt{\frac{1 - cos A}{1 + cos A}}$

Note:

These relations are very useful to find the trigonometric ratios of 22 ½°, 7 ½°, 11 ¼°, etc.

How to determine the signs of sin A/2, cos A/2 and tan A/2?

If A is given then we can easily find the quadrant in which A/2 lies.

Therefore, using the rule of “All, sin, tan, cos” we can find the exact signs of sin A/2, cos A/2 and tan A/2. In other words, if the value of cos A is given then A can have infinite number of values.

Hence, it is not possible to find the exact quadrant in which A/2 will lie.

Therefore, sin A/2, cos A/2 or tan A/2  may be positive as well as negative.  

● 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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