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受欢迎的三角函数 >

sin^3(x)+3sin^2(x)+2sin(x)=0,0<= x<2pi

解答

sin^{3} (x) + 3 sin^{2} (x) + 2 sin (x) = 0, 0 \leq x < 2 π

解答

x = 0, x = π, x = \frac{3 π}{2}

+1

度数

x = 0^{\circ}, x = 18 0^{\circ}, x = 27 0^{\circ}

求解步骤

sin^{3} (x) + 3 sin^{2} (x) + 2 sin (x) = 0, 0 \leq x < 2 π

用替代法求解

sin^{3} (x) + 3 sin^{2} (x) + 2 sin (x) = 0

令

: sin (x) = u

u^{3} + 3 u^{2} + 2 u = 0

u^{3} + 3 u^{2} + 2 u = 0 : u = 0, u = - 1, u = - 2

u^{3} + 3 u^{2} + 2 u = 0

因式分解

u^{3} + 3 u^{2} + 2 u : u (u + 1) (u + 2)

u^{3} + 3 u^{2} + 2 u

因式分解出通项

u : u (u^{2} + 3 u + 2)

u^{3} + 3 u^{2} + 2 u

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= u^{2} u + 3 uu + 2 u

因式分解出通项

u

= u (u^{2} + 3 u + 2)

= u (u^{2} + 3 u + 2)

分解

u^{2} + 3 u + 2 : (u + 1) (u + 2)

u^{2} + 3 u + 2

将表达式拆分成组

u^{2} + 3 u + 2

定义

2

的因数:

1, 2

2

约数 (因数)

找到

2

的质因数:

2

2

2

是质数，因此无法因数分解

= 2

加 1

1

2

的因数

1, 2

对于每两个因数

u * v = 2

，检验是否

u + v = 3

检验

u = 1, v = 2 : u * v = 2, u + v = 3 \Rightarrow

真

u = 1, v = 2

分组为

(a x^{2} + ux) + (vx + c)

(u^{2} + u) + (2 u + 2)

= (u^{2} + u) + (2 u + 2)

从

u^{2} + u

分解出因式

u

:

u (u + 1)

u^{2} + u

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= uu + u

因式分解出通项

u

= u (u + 1)

从

2 u + 2

分解出因式

2

:

2 (u + 1)

2 u + 2

因式分解出通项

2

= 2 (u + 1)

= u (u + 1) + 2 (u + 1)

因式分解出通项

u + 1

= (u + 1) (u + 2)

= u (u + 1) (u + 2)

u (u + 1) (u + 2) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or u + 1 = 0 or u + 2 = 0

解

u + 1 = 0 : u = - 1

u + 1 = 0

将

1

到右边

u + 1 = 0

两边减去

1

u + 1 - 1 = 0 - 1

化简

u = - 1

u = - 1

解

u + 2 = 0 : u = - 2

u + 2 = 0

将

2

到右边

u + 2 = 0

两边减去

2

u + 2 - 2 = 0 - 2

化简

u = - 2

u = - 2

解为

u = 0, u = - 1, u = - 2

u = sin (x)

代回

sin (x) = 0, sin (x) = - 1, sin (x) = - 2

sin (x) = 0, sin (x) = - 1, sin (x) = - 2

sin (x) = 0, 0 \leq x < 2 π : x = 0, x = π

sin (x) = 0, 0 \leq x < 2 π

sin (x) = 0

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

在

0 \leq x < 2 π

范围内的解

x = 0, x = π

sin (x) = - 1, 0 \leq x < 2 π : x = \frac{3 π}{2}

sin (x) = - 1, 0 \leq x < 2 π

sin (x) = - 1

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

在

0 \leq x < 2 π

范围内的解

x = \frac{3 π}{2}

sin (x) = - 2, 0 \leq x < 2 π :

无解

sin (x) = - 2, 0 \leq x < 2 π

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = 0, x = π, x = \frac{3 π}{2}

作图

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\frac{cos ( x )}{2} = \frac{- 3}{2}

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