Expert: Sherry Wallin - 12/16/2009

Question
Solve the Following Trigonometric Equations. Give All Positive Values of the angle between 0 degrees and 360 degrees that will satisfy each.

A. Cos 2x-sin^2 x/2 + 3/4 = 0

B. Sin^2(theta) = cos^2(theta) + 1/2

C. cos 4x = sin 2x

Im really not understanding this stuff....can you show me the answer and how you got it?


Answer
I will help you with C: Change cos(4x) into cos 2(2x) and use the identity: cos 2u= 1-2sin^2 u, so now you have 1-2sin^2(2x) = sin 2x -> 2sin^2(2x) + sin 2x -1 = 0 -> (2sin(2x) -1)(sin(2x) + 1) = 0 ->
sin(2x) = 1/2 -> well you know that the sin of 30 deg = 1/2 so wherever the sin x = 1/2 are the angles and that would be at 30 deg and at 150 deg. For the other factor solve it for the sin(2x) getting sin(2x) = -1 and where does the sin x = -1? When x = 270 deg but we want to know the sin (2x) so we want to know when 2x = 270 -> x = 135 and that is the only angle in the interval [0,2pi] that the sin x takes on that value.

Math Prof