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.
11 and 12 Grade Math
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