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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 cos2 A - sin2 A?

                                                      Or

How to proof the formula of cos 2A is equals 1 - 2 sin2 A?

                                                      Or

How to proof the formula of cos 2A is equals 2 cos2 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 = cos2 A - sin2 A

⇒ cos 2A = cos2 A - (1 - cos2 A), [since we know that sin2 θ = 1 - cos2 θ]

⇒ cos 2A = cos2 A - 1 + cos2 A,

cos 2A = 2 cos2 A - 1

⇒ cos 2A = 2 (1 - sin2 A) - 1, [since we know that cos2 θ = 1 - sin2 θ]

⇒ cos 2A = 2 - 2 sin2 A - 1

cos 2A = 1 - 2 sin2 A

Note:  

(i) From cos 2A = 2 cos2 A - 1 we get, 2 cos2 A = 1 + cos 2A

and from cos 2A = 1 - 2 sin2 A we get,  2 sin2A = 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° = cos2 60° - sin2 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)

= cos2 (2A) - sin2 (2A)


2. Express cos 4β in terms of sin 2β

Solution:

cos 4β

= cos (2 ∙ 2β)

= 1 - 2 sin2 (2β)


3. Express cos 4θ in terms of cos 2θ

Solution:

cos 4θ

= cos 2 ∙ 2θ

= 2 cos2 (2θ) – 1


4. Express cos 4A in term of cos A.

Solution:

cos 4A = cos (2 ∙ 2A) = 2 cos2 (2A) - 1

⇒ cos 4A = 2(2 cos 2A - 1)2 - 1

⇒ cos 4A = 2(4 cos4 A - 4 cos2 A + 1) - 1

⇒ cos 4A =  8 cos4 A – 8 cos2 A + 1


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

5. If sin A = 35 find the values of cos 2A.

Solution:

Given, sin A = 35

   cos 2A

= 1 - 2 sin2 A

= 1 - 2 (35)2

= 1 - 2 (925)

= 1 - 1825

= 251825

= 725


6. Prove that cos 4x = 1 - sin2 x cos2 x

Solution:

L.H.S. = cos 4x

= cos (2 × 2x)

= 1 - 2 sin2 2x, [Since, cos 2A = 1 - 2 sin2 A]

= 1 - 2 (2 sin x cos x)2

= 1 - 2 (4 sin2 x cos2 x)

= 1 - 8 sin2 x cos2 x = R.H.S.           Proved

 Multiple Angles








11 and 12 Grade Math

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