another trig question?

Question

solve step by step sin³Θ cosΘ -sinΘ cos³Θ = -1/4

Answer 1

First factor out sinOcosO:
sinOcosO(sin^2O - cos^2O) = -1/4

Multiply both sides by 2 to get an identity for sin 2O

2sinOcosO(sin^2O - cos^2O) = -1/2

Multiply both sides by -1 to reverse the part in parentheses and make it an identity for cos 2O

2 sinOcosO (cos^2O - sin^2O) = 1/2

Substitute the identities

sin 2O cos 2O = 1/2

Mult both by 2 to get another identity for sin 2(2O)

2 sin 2O cos 2O = 1

sin 4O = 1


Since sin pi/2 = 1, 4O = pi/2; therefore O = pi/8

Finally notice that this answer repeats at 5 pi/8, 9 pi/8, .... and reversing, -3 pi/8, -7 pi/8,...

So you could write the answers as (4n+1) pi / 8 where n is an integer

Answer 2

let me use x instead of Θ. Then,

sin^3 x cos x - sin x cos^3 x = -1/4
=> sin x cos x (sin^2 x - cos^2 x) = -1/4
=> sin 2x cos 2x = 1/2
=> sin 4x = 1
then, 4x = n.pi +[(-1)^n](pi/2)
and,
x = (1/4) { n.pi + [(-1)^n](pi/2)}
where n is an integer.

Hope this helps, I've liberally used the sin 2x and cos 2x expressions and the fact that sin (pi/2) =1
I have given the general solution of the equation above, for each integer n, you will get a seperate solution for the equation. n=0 gives the principle value pi/2 rad. = 90 degrees

Answer 3

I am replacing Θby x
sin³Θ cosΘ -sinΘ cos³Θ = -1/4
-sinxcosx(cos^2x-sin^2x)=-1/4
-sinxcosxcos2x=-1/4
using(cos^2x-sin^2x)=cos2x
now multiply both sides by 2
-2sinxcosxcos2x=-2/4
-sin2xcos2x=-1/2
using2sinxcosx=sin2x
again multiply both sides by 2
-2sin2xcos2x=-2/2
-sin4x=-1
sin4x=1
4x=90degree
x=90/4degree