SOLUTION: Sin^4 2theta -2 sin^2 2theta = -1 (3tan 3theta)^2-27=0 Solve for theta. Solutions should have exact radian measures such that 0<theta<2pi. (< is equal to or greater than)

Algebra ->  Trigonometry-basics -> SOLUTION: Sin^4 2theta -2 sin^2 2theta = -1 (3tan 3theta)^2-27=0 Solve for theta. Solutions should have exact radian measures such that 0<theta<2pi. (< is equal to or greater than)      Log On


   

Question 1125838: Sin^4 2theta -2 sin^2 2theta = -1
(3tan 3theta)^2-27=0
Solve for theta. Solutions should have exact radian measures such that 0 (< is equal to or greater than)
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


First equation....

sin%282x%29%5E4-2%2Asin%282x%29%5E2%2B1+=+0
%28sin%282x%29%5E2-1%29%5E2+=+0
sin%282x%29%5E2-1+=+0
%28sin%282x%29-1%29%28sin%282x%29%2B1%29+=+0
sin%282x%29+=+1 or sin%282x%29+=+-1

0 < x < 2pi --> 0 < 2x < 4pi

On (0,4pi), sin(x) is 1 or -1 at pi/2, 3pi/2, 5pi/2, and 7pi/2.

On (0,2pi), sin(2x) is 1 or -1 at pi/2, 3pi/2, 5pi/2, and 7pi/2.

So the solution set for this equation, for x on (0,2pi), is

{pi/4, 3pi/4, 5pi/4, 7pi/4}

Second equation....

%283tan%283x%29%29%5E2+=+27
9%28tan%283x%29%29%5E2+=+27
%28tan%283x%29%29%5E2+=+3
tan%283x%29+=+sqrt%283%29 or tan%283x%29+=+-sqrt%283%29

0 < x < 2pi --> 0 < 3x < 6pi

On (0,6pi), tan(x) is sqrt(3) or -sqrt(3) at pi/3, 2pi/3, 4pi/3, 5pi/3, 7pi/3, 8pi/3, 10pi/3, 11pi/3, 13pi/3, 14pi/3, 16pi/3, and 17pi/3.

On (0,2pi), tan(3x) is sqrt(3) or -sqrt(3) at pi/9, 2pi/9, 4pi/9, 5pi/9, 7pi/9, 8pi/9, 10pi/9, 11pi/9, 13pi/9, 14pi/9, 16pi/9, and 17pi/9.

So the solution set for this equation, for x on (0,2pi), is

{pi/9, 2pi/9, 4pi/9, 5pi/9, 7pi/9, 8pi/9, 10pi/9, 11pi/9, 13pi/9, 14pi/9, 16pi/9, 17pi/9}