I need helpfinding the solutions to these trigonometricequations: (3)tan(x)sin(x

Question

I need helpfinding the solutions to these trigonometricequations: (3)tan(x)sin(x)+sin(x)=0
csc(3x)=sin(3x)
csc(x/2)=sin(x/2)
(answers should be in the form + 2k, 0 < 2)
2cos^2(x)-sin(x)=1 (findsolutions in the interval [0, 2))
3tan(x)=14 (a) Find solutions inthe form + k, 0 < 2) (b) Use acalculator to solve the equation in the interval [0,2)
6sin(2x)-cos(x)=0 (a) Find allsolutions in theform + k or +2k, 0 < 2 (b) Use acalculator to solve the equation in the interval [0,2)
ThankYou! I need helpfinding the solutions to these trigonometricequations: (3)tan(x)sin(x)+sin(x)=0
csc(3x)=sin(3x)
csc(x/2)=sin(x/2)
(answers should be in the form + 2k, 0 < 2)
2cos^2(x)-sin(x)=1 (findsolutions in the interval [0, 2))
3tan(x)=14 (a) Find solutions inthe form + k, 0 < 2) (b) Use acalculator to solve the equation in the interval [0,2)
6sin(2x)-cos(x)=0 (a) Find allsolutions in theform + k or +2k, 0 < 2 (b) Use acalculator to solve the equation in the interval [0,2)
ThankYou! (answers should be in the form + 2k, 0 < 2)
2cos^2(x)-sin(x)=1 (findsolutions in the interval [0, 2))
3tan(x)=14 (a) Find solutions inthe form + k, 0 < 2) (b) Use acalculator to solve the equation in the interval [0,2)
6sin(2x)-cos(x)=0 (a) Find allsolutions in theform + k or +2k, 0 < 2 (b) Use acalculator to solve the equation in the interval [0,2)
ThankYou!

Explanation / Answer

Bring stuff on the other side of theequation:

cos^2 x = 1 - sin^2 x
2cos^2 x - sin x - 1 = 0
2(1 - sin^2 x) - sin x - 1 = 0
2sin^2 x + sin x - 1 = 0

Factor this
(2 sin x - 1)(sin x +1) = 0
2 sin x - 1 = 0 or sin x +1 = 0
sin x = 1, and sin x = -1

Answers:
pi/6, and 3pi/2, adding 2pi times a constant, for each trip aroundthe unit circle.

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