Could anyone help me in trigonometry?

Question

Help me find all the angles from 0˚ to 360˚ inclusive which satsify the equations.
a. 2 sin^4 z + 7 cos^2 z = 4
b. 3 sin y tan y + 8 = 0
c. 2 sin 2x + 1 = 0
d. sec y(1+ tan y) = 6 cosec y
e. 4 sin y cos y - 3 cos^2 y = 0
f. sec (3z/2 - 18˚) + 2 = 0

I'll appreciate everyone's help, even if you help me at least a question.

Answer 1

[Waay too much work! Anyway here are some]

a. 2 sin^4 z + 7 cos^2 z = 4
Noting sin^2 z + cos^2 z =1... we try to complete the square:
2 sin^4 z + 7 cos^2 z - 7(sin^2 z + cos^2 z) +3 =0
2 sin^4 z - 7sin^2 z +3 =0
Solve as a quadratic in (sin^2 z):
sin^2 z = [ 7 +/- sqrt (49-24) ] / 4
sin^2 z = [7 +/-5 ] / 4 = 2/4 (or 3, extraneous root)
obviously sin^2 z = 3 doesn't have a solution
sin^2 z = 1/2 => sin z = 1/sqrt2
z = 45˚ or 135˚

b. 3 sin y tan y + 8 = 0
3 sin^2y/cos y = -8
3 (1-cos^2 y)/cos y = -8
Set c = cos y
(1-c^2)/c = -8/3
(1-c^2) = -8/3 c
c^2 - 8/3 c -1 = 0
c = [ 8/3 +/- sqrt ( 64/9 - 4 ) ] / 2
c = [ 8/3 - sqrt (28/9) ] / 2 ; (the + root is extraneous)
y = arccos (0.4514)
= 63.17˚ or 296.83˚

c. 2 sin 2x + 1 = 0
sin 2x = -1/2
2x = arcsin(-1/2) = 210˚ or 330˚ or 570˚ or 690˚ ...
x = 105˚ or 165˚ or 285˚ or 345˚

...
f. sec (3z/2 - 18˚) + 2 = 0
sec (3z/2 - 18˚) = -2
cos (3z/2 - 18˚) = -1/2
3z/2 - 18˚ = arccos(-1/2) = 120˚ or 240˚ or 480˚ or 600˚
3z/2 = 138˚ or 258˚ (or 498˚ or 618˚)
z = 92˚ or 172˚

Answer 2

c. 2 sin 2x + 1 = 0
sin 2x= -1/2
->2x = sin inverse(-1/2)
->2x = -30 degree
->x= -15 degree

Answer 3

a.2sin^4z+7(1-sin^2z)=4
2sin^4z+7-7sin^2z=4
2sin^4z-7sin^2z+3=0
2sin^4z-6sin^2z-sin^2z+3=0
2sin^2z(sin^2z-3)-1(sin^2z-3)=0
(sin^2z-3)(2sin^2z-1)=0
sin^2z=3
sinz=+/-sq.rt.3 not admissible as -1<=sinz<=1
2sin^2z=1
sin^2z=1/2
sinz=+/-1/sq.rt2
=pi/4 or 5pi/4,3pi/4,7pi/4

b.3sinytany+8=0
3siny*siny/cosy+8=0
3sin^2y+8cosy=0
3(1-cos^2y)+8cosy=0
3-3cos^2y+8cosy=0
3cos^2y-8cosy-3=0
3cos^2y-9cosy+cosy-3=0
3cosy(cosy-3)+1(cosy-3)=0
(3cosy+1)(cosy-3)=0
cosy=-1/3 y=arccos(-1/3)
cosy=3 not admissible as -1<=cosy<=1

c.2sin2x+1=0
2sin2x=-1
sin2x=-1/2
sin2x=sin7pi/6 or sin 11pi/6
2x=7pi/6 or 11pi/6
x=7pi/12 or 11pi/12

d.secy(1+tany)=6cosecy
1/cosy+siny/cos^2y= 6/siny
multiplying by sinycos^2y
sinycosy+sin^2y=6cos^2y
sinycosy+1-cos^2y=6cos^2y
sinycosy-7cos^2y+sin^2y+cos^2y=0
sin^2y+sinycosy-6cos^2y=0
sin^2y+3sinycosy-2sinycosy-6cos^2y=0
siny(siny+3cosy)-2cosy(siny+3cosy)=0
(siny-2cosy)(siny+3cosy)=0
siny=2cosy or tany=2
siny=-3cosyor tany=-3

e.cosy(4siny-3cosy)=0
cosy=o y=pi/2
4siny=3cosy or tany=3/4