Use double angle formula, addition formula for the sine and other relations amongst trigonometric funtions?

Question

Find an expression for sin(3x) in terms of sinx alone.

Use the expression found toshow that, if sinx = sin(3x), then either sinx -0 or sinx = ± 1/√2

Answer 1

using sin (A+B) rule, subbing A = x and B = 2x, you can work out sin3x = 3sin (x) - (4 sin x)^3

Answer 2

sin3x=  sin(2x+x)

        =  sin2x.cosx + cos2x.sinx

        =  2sinx.cosx.cosx + (1- 2sin^x).sinx

        =  2sinx( 1- sin^2x) + sinx - 2 sin^3x

        = 2sinx - 2sin^3x + sinx - 2sin^3x

        = 3sinx - 4sin^3x

for second part:

sinx = sin (3x)

or, sinx = 3sinx - 4sin^3x

or, 4sin^3x -2sinx =0

or 2sinx ( 2sin^x -1) =0

therefore:

either 2sinx = 0 ==> sinx = 0

or, 2sin^2x - 1= 0 or,sin^2x = 1/2 or, sinx = ± 1/√2

hope this helps mate!

Answer 3

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