Solve For All Values of X

Question

Solve for ALL values of x in the real number system.  If an EXACT solution does not exist, round your answers to the nearest hundredth.

 

sin2x = sinx

Answer 1

sin2x = sinx

 

use the identity: sin2x=2sinxcosx

 

2sinxcosx = sinx

 

2sinxcosx - sinx = 0

 

sinx(2cosx-1) = 0

 

sinx = 0  OR   2cosx-1=0

 

sin x = 0 when x is an integer multiple of pi, or x = npi (n = 0, +-1, +-2, etc)

 

2cosx-1=0

 

2cosx=1

 

cosx=1/2

 

cosx=1/2 when x = pi/3 + an integer multiple of 2pi ( x = pi/3 + 2npi)  or x = 5pi/3 + integer multiple of pi ( x = 5pi/3 + 2npi)

 

so our solution is:

 

x = npi, pi/3 + 2npi, 5pi/3 + 2npi (where n is any integer)