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