Sin3x formula derivation

Do you know sin3x formula? In trigonometry we have so many formulas to remember. Entire trigonometry we have formulas. In that sin3x formula is one of the important formulas.

Sin3x formula

Now we will learn the methods for deriving the formula for sin3x. Now we will find out the formula in terms of sinx and also in terms of sinx and cosx. Learn both the formulas. Based on your problem conditions you can use the required formula. But generally the formula will be in terms of sinx. This only better to remember. If you want in terms of sinx and cosx later you can modify it.

Derivation of sin3x formula:

Sin3x     = sin (2x+x)

               = sin2x cosx + sinx cos2x

               = 2 sinx cosx cosx + sinx ( 1-2sin2x)

               = 2 sinx cos2x + sinx – 2sin3x

               = 2 sinx (1 – sin2x) + sinx – 2 sin3x

               = 2 sinx –2 sin2x sinx + sinx –2 sin3x

               = 3 sinx – 4 sin3x

Therefore,

Sin3x = 3 sinx – 4 sin3x.

Sin3x formula in terms of sinx and cosx:

Sin3x     = sin (2x+x)

               = sin2x cosx + sinx cos2x

               = 2 sinx cosx cosx + sinx ( 2cos2x – 1)

               = 2 sinx cos2x + sinx 2 cos2x – sinx

               = 4 cos2x sinx – sinx

               = ( 4 cos2x – 1 ) sinx

Or

Sin3x     = sin (2x+x)

               = sin2x cosx + sinx cos2x

               = 2 sinx cosx cosx + sinx ( cos2x – sin2x)

               = 2 sinx cos2x + sinx cos2x – sin3x

               = 3 sinx cos2x – sin3x

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