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

● Multiple Angles

• sin 2A in Terms of A
• cos 2A in Terms of A
• tan 2A in Terms of A
• sin 2A in Terms of tan A
• cos 2A in Terms of tan A
• Trigonometric Functions of A in Terms of cos 2A
  
• sin 3A in Terms of A
• cos 3A in Terms of A
• tan 3A in Terms of A
• Multiple Angle Formulae

11 and 12 Grade Math

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