F.4 附加數學 - 三角方程之通解

1) 2sin2x = 2cos2x﹣1
cos2x = 3/4
cosx = ±√(3/4)
cosx = ±√3/2
x = 180n°± 30°

2)4cos2θ﹣8sin θ + 1 = 0
4﹣4sin2θ﹣8sin θ + 1 = 0
4sin2θ + 8sin θ﹣5 = 0
(2sinθ + 5)(2sinθ﹣1) = 0
sinθ = -5/2 (捨去)或sinθ = 1/2
∴θ = 180n°+ (-1)n30°
3)4sin2x + 3cos x﹣3 = 0
4﹣4cos2x + 3cos x﹣3 = 0
4cos2x﹣3cos x﹣1 = 0
(4cosx + 1)(cosx﹣1) = 0
cosx = -1/4或cosx = 1
∴x = 360n°± 104.5°或 x = 360n°

4)tan2θ = 1 + sec θ
sec2θ﹣1 = 1 + sec θ
sec2θ﹣sec θ﹣2 = 0
(secθ﹣2)(secθ + 1) = 0
secθ = 2或secθ = -1
cosθ = 1/2或cosθ = -1
∴θ = 360n°±60°或 x = 360n°±180°

5)sin θ = 2 tan θ
sin θ = 2sinθ / cosθ
sinθ cosθ = 2sinθ
sinθ cosθ﹣2sinθ = 0
sinθ(cosθ﹣2) = 0
sinθ = 0或cosθ = 2(捨去)
∴θ = 180n°

6)sin5x = cos3x
sin5x﹣sin(90°﹣3x) = 0
2sin(4x﹣45°) cos(45°+ x) = 0
cos(45°+ x) = 0或sin(4x﹣45°) = 0
45°+ x = 360n°±90°或 4x﹣45°= 180n°
∴x = 360n°+ 45°或x = 360n°﹣135°或x = 45n°+ 11.25°
或相等地，
x = 180n°+ 45°或x = 45n°+ 11.25°

1)2sin^2x=2cos^2x-1
2sin^2x=2-2sin^2x-1
sinx=1/2 or -1/2
x=30or150
general solution:x=180n+(-1)^n30
or x=180n+(-1)^n150.
2-5 is so easy to solve, you can do it yourself.

6)sin5x=cos3x
sin5x=sin(90-3x)
5x=90-3x or 5x=3x-90
x=11.25 or x=-45
general solution:
x=180n+(-1)^n(11.25)or x=180n-(-1)^n(45)