trigonometry Q4

1)2sinx = tanx
2sinx = sinx / cosx
2sinxcosx = sinx

sinx (2cosx ﹣1) = 0

sinx = 0 or cosx = 1 / 2

x = 0, 180, 360 or x = 60, 300
∴x = 0,60,180,300,360

2)2cosx = 3tanx
2cosx = 3(sinx/cosx)
2cos2x = 3sinx
2(1 ﹣sin2x) = 3sinx
2 ﹣2sin2x = 3sinx
2sin2x ﹣3sinx ﹣2 = 0
(2sinx + 1)(sinx ﹣2) = 0
2sinx + 1 = 0 or sinx ﹣2 = 0
sinx = -1/2 or sinx = 2 (rejected,-1 <= sinx <= 1)
x = 210,330


3)L.H.S.
= tan2 x ﹣1 ﹣3cos2 x
= (sin2 x / cos2 x) ﹣1 ﹣3cos2 x
= (sin2 x / cos2 x) ﹣(cos2 x/cos2 x) ﹣[3cos2 x(cos2 x)/cos2 x]
= (sin2 x ﹣cos2 x ﹣3cos4 x) / cos2 x
= [(1 ﹣cos2 x) ﹣cos2 x ﹣3cos4 x] / cos2 x
= (1 ﹣2cos2 x ﹣3cos4 x) / cos2 x
= R.H.S.

4)3cos2 x = tan2 x ﹣1
3cos2 x = sec2 x ﹣2
3cos2 x = (1 / cos2 x) ﹣2
3cos4 x = 1 ﹣2cos2 x
3cos4 x + 2cos2 x ﹣1 = 0
(3cos2 x ﹣1)(cos2 x + 1) = 0
cos2 x = 1/2 or cos2 x = -1(rejected)
cosx = √1/2 or -√1/2
x = 60,120,240,300

2008-04-20 21:33:44 補充：
tan^2 x + 1 = sec^2 x
tan^2 x = sec^2 x ﹣1
tan^2 x ﹣1 = sec^2 x ﹣2