How do you use the Euler's formula to obtain the sin3X in terms of cosX and sinX?

Answer 1

Using Euler's Formula, you use

(cos(x) + i sin(x))^n = cos (nx) + i sin(nx)

Now you let n=3

(cos(x) + i sin (x))3 = cos(3x) + i sin (3x)

(cos(x))3 + 3(cos(x))2 * i sin(x) + 3cos(x) * i2 (sin(x))3 = cos(3x)+ i sin(3x)

(cos(x))3 + i(3sin(x)(cos (x))2) - 3cos(x)(sin(x)2) - i(sin(x))3 = cos (3x) + i sin(3x)

Now only use the terms with i in them to figure out what sin(3x) is...

3sin(x)(cos(x))2 - (sin(x))3 = sin(3x)

Hope this helps! :D