Step by step solution to "How to express sin(3 theta) in terms of..."

kejuuoo

Answered question

2023-01-07

How to express sin(3 theta) in terms of trigonometric functions of theta?

haushHubalth3c

Beginner2023-01-08Added 5 answers

Finding the required formula:
As you know,

\sin (x + y) = \sin x \cos y + \cos x \sin y \cos (x + y) = \cos x \cos y - \sin x \sin y

We can write that

\sin (3 θ) = \sin (2 θ + θ) = \sin (2 θ) \cos (θ) + \cos (2 θ) \sin (θ) = \sin (θ + θ) \cos (θ) + \cos (θ + θ) \sin (θ) = [\sin (θ) \cos (θ) + \cos (θ) \sin (θ)] \cos (θ) + [\cos (θ) \cos (θ) + \sin (θ) \sin (θ)] \sin (θ) = 2 \sin (θ) \cos^{2} (θ) + \sin (θ) \cos^{2} (θ) - \sin^{3} (θ) = 3 \sin (θ) \cos^{2} (θ) - \sin^{3} (θ)

= 3 \sin (θ) (1 - \sin^{2} (θ)) - \sin^{3} (θ) = 3 \sin (θ) - 3 \sin^{3} (θ) - \sin^{3} (θ) = 3 \sin (θ) - 4 \sin^{3} (θ)

(\sin^{2} θ + \cos^{2} θ = 1)

Therefore, the value of

\sin (3 θ)

3 \sin (θ) - 4 \sin^{3} (θ)

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?