Question

Express sin^5 x as a linear combination of sin kx and coskx, k belongs to Z

Answer 1

Express sin⁵ (x) as a linear combination of sin (kx) and cos (kx)

sin⁵ (x) = sin³ (x) * sin² (x)

We use the degree reduction formula:

sin² (x) = (1 - cos (2x))/2

sin³ (x) = (3*sin (x) − sin (3x))/4

We get:

sin⁵ (x) = (3*sin (x) − sin (3x)) * (1 − cos (2x)) / 8 =

= (3*sin (x) − sin (3x) - 3*sin (x) * cos (2x) + sin (3x) * cos (2x))/8

We use the formula for the product of sines and cosines:

sin (a) * cos (b) = (sin (a+b) + sin (a-b))/2

sin(-a) = - sin(a)

We get:

sin⁵ (x) = (6*sin (x) − 2*sin (3x) - 3*sin (3x) + 3*sin (x) + sin (5x) + sin (x))/16 =

= (sin (5x) - 5*sin (3x) + 10*sin (x))/16

Answer: sin⁵ (x) = (sin (5x) - 5*sin (3x) + 10*sin (x))/16