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受欢迎的三角函数 >

sin^3(x)=2sin(x)

解答

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

解答

x = 2 πn, x = π + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: sin (x) = u

u^{3} = 2 u

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

u^{3} = 2 u

将

2 u

para o lado esquerdo

u^{3} = 2 u

两边减去

2 u

u^{3} - 2 u = 2 u - 2 u

化简

u^{3} - 2 u = 0

u^{3} - 2 u = 0

因式分解

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

u^{3} - 2 u

因式分解出通项

u : u (u^{2} - 2)

u^{3} - 2 u

使用指数法则:

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

u^{3} = u^{2} u

= u^{2} u - 2 u

因式分解出通项

u

= u (u^{2} - 2)

= u (u^{2} - 2)

分解

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

u^{2} - 2

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

= (u + 2) (u - 2)

= u (u + 2) (u - 2)

u (u + 2) (u - 2) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or u + 2 = 0 or u - 2 = 0

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

u - 2 = 0 : u = 2

u - 2 = 0

将

2

到右边

u - 2 = 0

两边加上

2

u - 2 + 2 = 0 + 2

化简

u = 2

u = 2

解为

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

u = sin (x)

代回

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

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

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

sin (x) = - 2 :

无解

sin (x) = - 2

- 1 \leq sin (x) \leq 1

无解

sin (x) = 2 :

无解

sin (x) = 2

- 1 \leq sin (x) \leq 1

无解

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

tan (x) = (\frac{3}{2})

2sin(3x)=1,0<= x<= 2pi

2 sin (3 x) = 1, 0 \leq x \leq 2 π

3cot(x)=2sin(x)

3 cot (x) = 2 sin (x)

tan (a) = \frac{7}{12}

sin(5x)=cos(3x)

sin (5 x) = cos (3 x)