We will learn how to express the multiple angle of cos 3A in terms of A or cos 3A in terms of cos A.
Trigonometric function of cos 3A in terms of cos A is also known as one of the double angle formula.
If A is a number or angle then we have, cos 3A = 4 cos^3 A - 3 cos A
Now we will proof the above multiple angle formula step-by-step.
Proof: cos 3A
= cos (2A + A)
= cos 2A cos A - sin 2A sin A
= (2 cos^2 A - 1) cos A - 2 sin A cos A ∙ sin A
= 2 cos^3 A - cos A - 2 cos A (1 - cos^2 A)
= 2 cos^3 A - cos A - 2 cos A + 2 cos^3 A
= 4 cos^3 A - 3 cos A
Therefore, cos 3A = 4 cos^3 A - 3 cos A Proved
Note: (i) In
the above formula we should note that the angle on the R.H.S. of the
formula is one-third of the angle on L.H.S. Therefore, cos 120° = 4 cos^3 40° - 3 cos 40°.
(ii) To find the formula of cos 3A in terms of A or cos 3A in terms of cos A we have use cos 2A = 2cos^2 A - 1.
Now, we will apply the formula of multiple angle of cos 3A in terms of A or cos 3A in terms of cos A to solve the below problems.
1. Prove that: cos 6A = 32 cos^6 A - 48 cos^4 A + 18 cos^2 A - 1
Solution:
L.H.S. = cos 6A
= 2 cos^2 3A - 1, [Since we know that, cos 2θ = 2 cos^2 θ - 1]
= 2(4 cos^3 A - 3 cos A)^2 - 1
= 2 (16 cos^ 6 A + 9 cos^2 A - 24 cos^2 A) - 1
= 32 cos^6 A – 48 cos^4 A + 18 cos^2 A - 1 = R.H.S.
2. Show that, 32 sin^6 θ = 10 - 15 cos 2θ + 6 cos 4θ - cos 6θ
Solution:
L.H.S = 32 sin^6 θ
= 4 ∙ (2 sin^2 θ)^3
= 4 (1 - cos 2θ)^3
= 4 [1 - 3 cos 2θ + 3 ∙ cos^2 2θ - cos^3 2θ]
= 4 - 12 cos^2 θ + 12 cos^2 2θ - 4 cos^3 2θ
= 4 - 12 cos 2θ + 6 ∙ 2 cos^2 2θ - [cos 3 ∙ (2θ) + 3 cos 2θ]
[Since, cos 3A = 4 cos^3 A - 3 cos A
Therefore, 4 cos^3 A = cos 3A + 3 cos A]
⇒ 4 cos^3 2θ = cos 3 ∙ (2θ) + 3 cos 2θ, (replacing A by 2θ)
= 4 - 12 cos 2θ + 6 (1 + cos 4θ) - cos 6θ - 3 cos 2θ
= 10 - 15 cos 2θ + 6 cos 4θ - cos 6θ = R.H.S. Proved
3. Prove that: cos A cos (60 - A) cos (60 + A) = ¼ cos 3A
Solution:
L.H.S. = cos A ∙ cos (60 - A) cos (60 + A)
= cos A ∙ (cos^2 60 - sin^2 A), [Since we know that cos (A + B) cos (A - B) = cos ^2 A - sin ^2 B]
= cos A (¼ - sin^2 A)
= cos A (¼ - (1 - cos^2 A))
= cos A (-3/4 + cos ^2 A)
= ¼ cos A (-3 + 4 cos^2 A)
= ¼(4 cos^3A - 3 cos A)
= ¼ cos 3A = R.H.S. Proved
11 and 12 Grade Math
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