cos 2A in Terms of A

We will learn to express trigonometric function of cos 2A in terms of A. We know if A is a given angle then 2A is known as multiple angles.

How to proof the formula of cos 2A is equals cos$^{2}$ A - sin$^{2}$ A?

                                                      Or

How to proof the formula of cos 2A is equals 1 - 2 sin$^{2}$ A?

                                                      Or

How to proof the formula of cos 2A is equals 2 cos$^{2}$ A - 1?

We know that for two real numbers or angles A and B,

cos (A + B) = cos A cos B - sin A sin B

Now, putting B = A on both sides of the above formula we get,

cos (A + A) = cos A cos A - sin A sin A

⇒ cos 2A = cos$^{2}$ A - sin$^{2}$ A

⇒ cos 2A = cos$^{2}$ A - (1 - cos$^{2}$ A), [since we know that sin$^{2}$ θ = 1 - cos$^{2}$ θ]

⇒ cos 2A = cos$^{2}$ A - 1 + cos$^{2}$ A,

⇒ cos 2A = 2 cos$^{2}$ A - 1

⇒ cos 2A = 2 (1 - sin$^{2}$ A) - 1, [since we know that cos$^{2}$ θ = 1 - sin$^{2}$ θ]

⇒ cos 2A = 2 - 2 sin$^{2}$ A - 1

⇒ cos 2A = 1 - 2 sin$^{2}$ A

Note:  

(i) From cos 2A = 2 cos$^{2}$ A - 1 we get, 2 cos$^{2}$ A = 1 + cos 2A

and from cos 2A = 1 - 2 sin$^{2}$ A we get,  2 sin$^{2}$A = 1 - cos 2A

(ii) In the above formula we should note that the angle on the R.H.S. is half of the angle on L.H.S. Therefore, cos 120° = cos$^{2}$ 60° - sin$^{2}$ 60°.

(iii) The above formulae is also known as double angle formulae for cos 2A.

Now, we will apply the formula of multiple angle of cos 2A in terms of A to solve the below problems.

1. Express cos 4A in terms of sin 2A and cos 2A

Solution:

cos 4A

= cos (2 ∙ 2A)

= cos$^{2}$ (2A) - sin$^{2}$ (2A)

2. Express cos 4β in terms of sin 2β

Solution:

cos 4β

= cos (2 ∙ 2β)

= 1 - 2 sin$^{2}$ (2β)

3. Express cos 4θ in terms of cos 2θ

Solution:

cos 4θ

= cos 2 ∙ 2θ

= 2 cos$^{2}$ (2θ) – 1

4. Express cos 4A in term of cos A.

Solution:

cos 4A = cos (2 ∙ 2A) = 2 cos$^{2}$ (2A) - 1

⇒ cos 4A = 2(2 cos 2A - 1)$^{2}$ - 1

⇒ cos 4A = 2(4 cos$^{4}$ A - 4 cos$^{2}$ A + 1) - 1

⇒ cos 4A =  8 cos$^{4}$ A – 8 cos$^{2}$ A + 1

More solved examples on cos 2A in terms of A.

5. If sin A = $\frac{3}{5}$ find the values of cos 2A.

Solution:

Given, sin A = $\frac{3}{5}$

   cos 2A

= 1 - 2 sin$^{2}$ A

= 1 - 2 ($\frac{3}{5}$)$^{2}$

= 1 - 2 ($\frac{9}{25}$)

= 1 - $\frac{18}{25}$

= $\frac{25 - 18}{25}$

= $\frac{7}{25}$

6. Prove that cos 4x = 1 - sin$^{2}$ x cos$^{2}$ x

Solution:

L.H.S. = cos 4x

= cos (2 × 2x)

= 1 - 2 sin$^{2}$ 2x, [Since, cos 2A = 1 - 2 sin$^{2}$ A]

= 1 - 2 (2 sin x cos x)$^{2}$

= 1 - 2 (4 sin$^{2}$ x cos$^{2}$ x)

= 1 - 8 sin$^{2}$ x cos$^{2}$ x = R.H.S.           Proved

● 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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