Recall the derivation of a formula for #sin 3theta#:
#sin 3theta = sin(theta+2theta)#
#= sin theta * cos 2theta + sin 2theta * cos theta#
#= sin theta * (cos^2 theta-sin^2 theta) + 2sin theta*cos theta*cos theta#
#= sin theta * (1-2sin^2 theta) + 2sin theta * (1-sin^2 theta)#
#= 3sin theta - 4 sin^3 theta#
Applying this to our equation, we get
#3sin theta - 4 sin^3 theta = sin theta#
or
#2sin theta - 4sin^3 theta = 0#
or
#sin theta*(1-2sin^2 theta) = 0#
This equation has one set of solutions when #sin theta = 0#, that is
(Solution 1.1) #theta = 0+2pi n# and
Solution 1.2) #theta=pi+2pi n#,
which can be combined into
(Solution 1') #theta = pi n#, where #n# - any integer number.
Another set of solution is from
#1-2sin^2 theta = 0#
or
#sin theta = +-sqrt(2)/2#
With "#+#" sign the solutions are
(Solution 2.1) #theta = pi/4 +2pi n# and
(Solution 2.2) #theta = (3pi)/4+2pi n#
With "#-#" sign the solutions are
(Solution 3.1) #theta = -pi/4+2pi n# and
(Solution 3.2) #theta = -(3pi)/4 + 2pi n#
In both cases #n# is any integer number.
NOTE: Solutions 2.1, 2.2, 3.1 and 3.2 can be combined into one expression: #theta = pi/4+pi/2n#, where #n# - any integer number.
CHECK (we can ignore #2pi n# since #2pi# is a period for all participating functions)
Solution 1.1:
Left side equals
#sin (3*0) = 0#
Right side equals
#sin 0 = 0#
Solution 1.2:
Left side equals
#sin (3pi) =# [since #2pi# is a period] #= sin (3pi-2pi) = sin(pi) = 0#
Right side equals
#sin(pi) = 0#
Solution 2.1:
Left side equals
#sin (3*pi/4) = sin (pi-pi/4) = # [since #sin phi=sin(pi-phi)#] #= sin(pi/4) = sqrt(2)/2#
Right side equals
#sin(pi/4) = sqrt(2)/2#
Solution 2.2:
Left side equals
#sin (3*(3pi)/4) = sin ((9pi)/4)=sin((9pi)/4-2pi) = sin(pi/4) = sqrt(2)/2#
Right side equals
#sin ((3pi)/4) = sqrt(2)/2# (see 2.1 above)
Solution 3.1:
Left side equals
#sin (3*(-pi)/4) = #[since #sin(-phi)=-sin(phi)#] #= -sin ((3pi)/4) = - sqrt(2)/2#
Right side equals
#sin((-pi)/4) = #[since #sin(-phi)=-sin(phi)#] #=-sqrt(2)/2#
Solution 3.2:
Left side equals
#sin (3*(-3pi)/4) = sin ((-9pi)/4)=sin((-9pi)/4+2pi) = sin(-pi/4) = -sqrt(2)/2#
Right side equals
#sin ((-3pi)/4) = -sin((3pi)/4) = -sqrt(2)/2#
