How do You Express sin 3 theta in Terms of Trigonometric Functions of theta

Answer: Sin3 theta gives the value of the sine trigonometric function for triple angle.

Sin3 theta is equal to 3 sin theta – 4 sin³ theta which can be proved by the angle addition formula of the sine function.

Following trigonometric identities will be used to prove the sin3 theta identity:

      sin²x + cos²x = 1

      sin (a+b) = sin a cos b + cos a sin b

      sin 2x = 2sinxcosx

Using angle addition formula:

sin3x = sin (2x + x)

= sin²x cosx + cos²x sinx 

= (2 sin x cos x) cos x + (1 – 2sin²x) sin x

= 2cos²x sin x – 2sin³x + sin x

= 2 (1 – sin²x) sin x – 2sin³x + sin x 

= 2 sin x – 2sin³x – 2sin³x + sin x

= 2 sin x + sin x – 2sin³x – 2sin³x

Which implies that sin3x =  3 sin x – 4 sin³x (Hence proved)

Answer: sin 3 theta can be expressed as 3 sin theta – 4 sin³ theta