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.

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**11 and 12 Grade Math**

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