SOLUTION: Solve the equation by factoring. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Solve the equation by factoring. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution      Log On


   

Question 1009675: Solve the equation by factoring. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
4 sin(θ) cos(θ) + 2 sqrt(3) sin(θ) − 2 sqrt(3) cos(θ) − 3 = 0
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
THE EXPECTED SOLUTION:
I do not know how you are expected to solve this.
No idea what is meant by "Round terms to three decimal places where appropriate", either.
Are you expected to round sqrt%283%29 or 2sqrt%283%29 to 3-decimal places? Are you expected to find the solution values for theta in radians rounded to 3-decimal places? In degrees?

MY SOLUTION:
I will temporarily rename variables, so as not to have to keep writing sin%28theta%29 and cos%28theta%29 so many times.
x=cos%28theta%29 and y=sin%28theta%29 .
With those changes, the equation looks like
4xy%2B2sqrt%283%29y-2sqrt%283%29x-3=0
Now I rearrange and factor by parts:
%284xy-2sqrt%283%29x%29%2B%282sqrt%283%29y-3%29=0
2x%282y-sqrt%283%29%29%2Bsqrt%283%29%282y-sqrt%283%29%29=0
%282x%2Bsqrt%283%29%29%282y-sqrt%283%29%29=0-->system%282x%2Bsqrt%283%29=0%2C%22or%22%2C2y-sqrt%283%29=0%29-->system%28x=-sqrt%283%29%2F2%2C%22or%22%2Cy=sqrt%283%29%2F2%29 .
So, going back to theta
system%28cos%28theta%29=-sqrt%283%29%2F2%2C%22or%22%2Csin%28theta%29=sqrt%283%29%2F2%29
In the interval %22%5B+0+%2C%222pi%22%5D%22 , the solutions fortheta (in radians) are
cos%28theta%29=-sqrt%283%29%2F2-->theta=pi+%2B-+pi%2F6 (rounds to 2.618 and 3.665).
sin%28theta%29=-sqrt%283%29%2F2-->theta=pi%2F2+%2B-+pi%2F6 (rounds to 1.047 and 2.094).