We will learn to express trigonometric function of sin 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 sin 2A is equals 2 sin A cos A?
We know that for two real numbers or angles A and B,
sin (A + B) = sin A cos B + cos A sin B
Now, putting B = A on both sides of the above formula we get,
sin (A + A) = sin A cos A + sin A cos A
⇒ sin 2A = 2 sin A cos A
Note: 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, sin 60° = 2 sin 30° cos 30°.
The above formula is also known as double angle formulae for sin 2A.
Now, we will apply the formula of multiple angle of sin 2A in terms of A to solve the below problems.
1. Express sin 8A in terms of sin 4A and cos 4A
Solution:
sin 8A
= sin (2 ∙ 4A)
= 2 sin 4A cos 4A, [Since, we know sin 2A = 2 sin A cos A]
2. If sin A = \(\frac{3}{5}\) find the values of sin 2A.
Solution:
Given, sin A = \(\frac{3}{5}\)
We know that, sin\(^{2}\) A + cos\(^{2}\) A = 1
cos\(^{2}\) A = 1 - sin\(^{2}\) A
cos\(^{2}\) A = 1 - (\(\frac{3}{5}\))\(^{2}\)
cos\(^{2}\) A = 1 - \(\frac{9}{25}\)
cos\(^{2}\) A = \(\frac{25 - 9}{25}\)
cos\(^{2}\) A = \(\frac{16}{25}\)
cos A = √\(\frac{16}{25}\)
cos A = \(\frac{4}{5}\)
sin 2A
= 2 sin A cos A
= 2 ∙ \(\frac{3}{5}\) ∙ \(\frac{4}{5}\)
= \(\frac{24}{25}\)
3. Prove that,16 cos \(\frac{2π}{15}\) cos \(\frac{4π}{15}\) cos \(\frac{8π}{15}\) \(\frac{16π}{15}\) = 1.
Solution:
Let, \(\frac{2π}{15}\) = θ
L.H.S = 16 cos \(\frac{2π}{15}\) cos \(\frac{4π}{15}\) cos \(\frac{8π}{15}\) \(\frac{16π}{15}\) = 1.
= 16 cos θ cos 2θ cos 4θ cos 8θ, [Since, θ = \(\frac{2π}{15}\)]
= \(\frac{8}{sin θ}\) (2 sin θ cos θ) cos 2θ cos 4θ cos 8θ
= \(\frac{4}{sin θ}\) (2 sin 2θ cos 2θ) cos 4θ cos 8θ
= \(\frac{2}{sin θ}\) (2 sin 4θ cos 4θ) cos 8θ
= \(\frac{1}{sin θ}\) (2 sin 8θ cos 8θ)
= \(\frac{1}{sin θ}\) ∙ sin 16θ
= \(\frac{1}{sin θ}\) ∙ sin (15θ + θ)
= \(\frac{1}{sin θ}\) ∙ sin (2π + θ), [Since, \(\frac{2π}{15}\) = θ ⇒15θ = 2π]
= \(\frac{1}{sin θ}\) ∙ sin (θ), [Since, sin (2π + θ) = sin θ]
= 1 = R.H.S. Proved
11 and 12 Grade Math
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