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.