Trigonometric Functions of A in Terms of cos 2A

We will learn how to express trigonometric functions of A in terms of cos 2A or trigonometric ratios of an angle A in terms of cos 2A.

We know the formula of cos 2A and now we will apply the formula to proof the below trigonometric ratio of multiple angle.

(i) Prove that: cos\(^{2}\) A = \(\frac{1 + cos 2A}{2}\)  i.e., cos A = ±\(\sqrt{\frac{1 + cos 2A}{2}}\)

We know that, cos 2A = 2 cos^2 A - 1

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

i.e., cos A = ±\(\sqrt{\frac{1 + cos 2A}{2}}\)


(ii) Prove that: sin\(^{2}\) A = \(\frac{1 - cos 2A}{2}\)  i.e., sin A = ±\(\sqrt{\frac{1 + cos 2A}{2}}\)

We know that, cos 2A = 1 - 2 sin^2 A

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

i.e., sin A = ±\(\sqrt{\frac{1 + cos 2A}{2}}\)


(iii) Prove that: tan\(^{2}\) A = \(\frac{1 - cos 2A}{1 + cos 2A}\)  i.e., tan A = ±\(\sqrt{\frac{1 - cos 2A}{1 + cos 2A}}\)

We know that, tan\(^{2}\) A = \(\frac{sin^{2} A}{cos^{2} A}\)

⇒ \(\frac{1 - cos 2A}{1 + cos 2A}\)  

i.e., tan A = ±\(\sqrt{\frac{1 - cos 2A}{1 + cos 2A}}\)





11 and 12 Grade Math

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