trigonometry (please double check this)
posted by anon .
Solve the following trig equations. give all the positive values of the angle between 0 degrees and 360 degrees that will satisfy each. give any approximate value to the nearest minute only.
1. sin2ƒÆ = (sqrt 3)/2
2. sin^2ƒÆ = cos^2ƒÆ + 1/2
3. sin 2x  cosx = 0
4. cos 4x = sin 2x
(((* stands for degrees)))
q1
sin2ƒÆ = (sqrt 3)/2
2ƒÆ = sin^1 (sqrt3/2)
= 60*, 120*, 420*, 480*
ƒÆ = 30*, 60*, 210* and 240*
(not sure of the rest)
q2
sin^2ƒÆ = cos^2ƒÆ + 1/2
sin^2 ƒÆ  cos^2 ƒÆ = 1/2 ...(1)
sin^2 ƒÆ + cos^2 ƒÆ = 1 ...(2)
(1) + (2),
2sin^2 ƒÆ = 3/2
sin^2 ƒÆ = 3/4
sin ƒÆ = +ã3/2
ƒÆ = 60*, 120*, 240* and 300*
q3
sin 2x  cosx = 0
2sinxcosx  cosx = 0
cos x (2sin x  1) = 0
cos x = 0 or sin x = 1/2
x = 90*, 270* or
x = 30* 150*
answer, x = 30*, 90*, 150* and 270*
q4
cos 4x = sin 2x
2(sin 2x)^2 + sin 2x  1 = 0
(2sin 2x  1)(sin 2x + 1) = 0
sin 2x = 1, 1/2
2x = 30*, 150*, 270*, 390*, 510*, 630*
x = 15*, 75*, 135*, 195*, 255* and 315*

""ƒÆ "" is supposed to be theta.

q1  correct
q2 is correct but here is a simpler way.
Let Ø = ƒÆ
sin^2 Ø = cos^2 Ø + 1/2
cos^2 Ø  sin^2 Ø = 1/2
cos 2Ø = 1/2
2Ø = 120° , 240° ...
Ø = 60° , 120° , 240° , 300°
q3 correct
q4 also correct
good job!