F.4 數學(trigonometry)

1)
2cosx - 3sinx = 3(cosx + sinx)
2cosx - 3sinx = 3cosx + 3sinx
6sinx = -cosx
6sinx/(6cosx) = -cosx/(6cosx)
tanx = -1/6
x = (180-9.5)°, (360-9.5)°
x = 170.5°, 350.5°


= = = = =
2)
0 ≤ x ≤ 360°
0 ≤ (3x+6°) ≤ 1086°

-2tan(3x+6°) = 1
tan(3x+6°) = -1/2
3x+6° = (180-26.6)°, (360-26.6)°, (540-26.6)°, (720-26.6)°, (900-26.6°), (1080-26.6)°
x = 49.1°, 109.1°, 169.1°, 229.1°,289.1°, 349.1°


= = = = =
3)
0 ≤ x ≤ 360°
0 ≤ 2x ≤ 720°

(√2)cos2x - 4cos(90°-2x) = 0
(√2)cos2x - 4sin2x = 0
4sin2x = (√2)cos2x
4sin2x/(4cos2x) = (√2)cos2x/(4cos2x)
tan2x = (√2)/4
2x = 19.47°, (180+19.47)°, (360+19.47)°, (540+19.47°)
x = 9.7°, 99.7°, 189.7°, 279.7°


= = = = =
4)
2cosx = 3tanx
2cosx = 3(sinx/cosx)
2cos²x = 3sinx
2(1 - sin²x) = 3sinx
2 - 2sin²x = 3sinx
2sin²x + 3sinx - 2 = 0
(2sinx - 1)(sinx + 2) = 0
sinx = 1/2 or sinx = -2 (rejected)
x = 30°, (180-30)°
x = 30°, 150°


= = = = =
5)
sin²x - 2sinxcosx - cos²x = 0
(sin²x - 2sinxcosx - cos²x)/cos²x = 0
tan²x - 2tanx - 1 = 0
tanx = {2 ± √[(-2)² - 4(1)(-1)]} / 2
tanx = (1+√2), (1-√2)
x = 67.5°, (180+67.5)°, (180-22.5)°, (360-22.5)°
x = 67.5°, 247.5°, 157.5°, 337.5°

2011-05-08 18:09:28 補充：
Alternative method for Q.5:

sin²x - 2sinxcosx + cos²x = 2cos²x
(sinx - cosx)² = 2cos²x
sinx - cosx = ±(√2)cosx
(sinx - cosx)/cosx = ±(√2)cosx/cosx
tanx - 1 = ±√2
tanx = (1 + √2), (1 - √2)
x = 67.5°, (180+67.5)°, (180-22.5)°, (360-22.5)°
x = 67.5°, 247.5°, 157.5°, 337.5° ..... (ans.)