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受欢迎的三角函数 >

4sin^2(x-pi/3)=3

解答

4 sin^{2} (x - \frac{π}{3}) = 3

解答

x = \frac{2 π}{3} + 2 πn, x = 2 πn + π, x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sin^{2} (x - \frac{π}{3}) = 3

用替代法求解

4 sin^{2} (x - \frac{π}{3}) = 3

令

: sin (x - \frac{π}{3}) = u

4 u^{2} = 3

4 u^{2} = 3 : u = \frac{3}{2}, u = - \frac{3}{2}

4 u^{2} = 3

两边除以

4

4 u^{2} = 3

两边除以

4

\frac{4 u ^{2}}{4} = \frac{3}{4}

化简

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

化简

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

u = sin (x - \frac{π}{3})

代回

sin (x - \frac{π}{3}) = \frac{3}{2}, sin (x - \frac{π}{3}) = - \frac{3}{2}

sin (x - \frac{π}{3}) = \frac{3}{2}, sin (x - \frac{π}{3}) = - \frac{3}{2}

sin (x - \frac{π}{3}) = \frac{3}{2} : x = \frac{2 π}{3} + 2 πn, x = 2 πn + π

sin (x - \frac{π}{3}) = \frac{3}{2}

sin (x - \frac{π}{3}) = \frac{3}{2}

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{π}{3} = \frac{π}{3} + 2 πn, x - \frac{π}{3} = \frac{2 π}{3} + 2 πn

x - \frac{π}{3} = \frac{π}{3} + 2 πn, x - \frac{π}{3} = \frac{2 π}{3} + 2 πn

解

x - \frac{π}{3} = \frac{π}{3} + 2 πn : x = \frac{2 π}{3} + 2 πn

x - \frac{π}{3} = \frac{π}{3} + 2 πn

将

\frac{π}{3}

到右边

x - \frac{π}{3} = \frac{π}{3} + 2 πn

两边加上

\frac{π}{3}

x - \frac{π}{3} + \frac{π}{3} = \frac{π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} = \frac{π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

同类项相加:

- \frac{π}{3} + \frac{π}{3} = 0

= x

化简

\frac{π}{3} + 2 πn + \frac{π}{3} : \frac{2 π}{3} + 2 πn

\frac{π}{3} + 2 πn + \frac{π}{3}

对同类项分组

= \frac{π}{3} + \frac{π}{3} + 2 πn

合并分式

\frac{π}{3} + \frac{π}{3} : \frac{2 π}{3}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + π}{3}

同类项相加:

π + π = 2 π

= \frac{2 π}{3}

= \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

解

x - \frac{π}{3} = \frac{2 π}{3} + 2 πn : x = 2 πn + π

x - \frac{π}{3} = \frac{2 π}{3} + 2 πn

将

\frac{π}{3}

到右边

x - \frac{π}{3} = \frac{2 π}{3} + 2 πn

两边加上

\frac{π}{3}

x - \frac{π}{3} + \frac{π}{3} = \frac{2 π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} = \frac{2 π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

同类项相加:

- \frac{π}{3} + \frac{π}{3} = 0

= x

化简

\frac{2 π}{3} + 2 πn + \frac{π}{3} : 2 πn + π

\frac{2 π}{3} + 2 πn + \frac{π}{3}

对同类项分组

= 2 πn + \frac{π}{3} + \frac{2 π}{3}

合并分式

\frac{π}{3} + \frac{2 π}{3} : π

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 2 π}{3}

同类项相加:

π + 2 π = 3 π

= \frac{3 π}{3}

数字相除:

\frac{3}{3} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = \frac{2 π}{3} + 2 πn, x = 2 πn + π

sin (x - \frac{π}{3}) = - \frac{3}{2} : x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

sin (x - \frac{π}{3}) = - \frac{3}{2}

sin (x - \frac{π}{3}) = - \frac{3}{2}

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{π}{3} = \frac{4 π}{3} + 2 πn, x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

x - \frac{π}{3} = \frac{4 π}{3} + 2 πn, x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

解

x - \frac{π}{3} = \frac{4 π}{3} + 2 πn : x = 2 πn + \frac{5 π}{3}

x - \frac{π}{3} = \frac{4 π}{3} + 2 πn

将

\frac{π}{3}

到右边

x - \frac{π}{3} = \frac{4 π}{3} + 2 πn

两边加上

\frac{π}{3}

x - \frac{π}{3} + \frac{π}{3} = \frac{4 π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} = \frac{4 π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

同类项相加:

- \frac{π}{3} + \frac{π}{3} = 0

= x

化简

\frac{4 π}{3} + 2 πn + \frac{π}{3} : 2 πn + \frac{5 π}{3}

\frac{4 π}{3} + 2 πn + \frac{π}{3}

对同类项分组

= 2 πn + \frac{π}{3} + \frac{4 π}{3}

合并分式

\frac{π}{3} + \frac{4 π}{3} : \frac{5 π}{3}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 4 π}{3}

同类项相加:

π + 4 π = 5 π

= \frac{5 π}{3}

= 2 πn + \frac{5 π}{3}

x = 2 πn + \frac{5 π}{3}

x = 2 πn + \frac{5 π}{3}

x = 2 πn + \frac{5 π}{3}

解

x - \frac{π}{3} = \frac{5 π}{3} + 2 πn : x = 2 πn + 2 π

x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

将

\frac{π}{3}

到右边

x - \frac{π}{3} = \frac{5 π}{3} + 2 πn

两边加上

\frac{π}{3}

x - \frac{π}{3} + \frac{π}{3} = \frac{5 π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} = \frac{5 π}{3} + 2 πn + \frac{π}{3}

化简

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

同类项相加:

- \frac{π}{3} + \frac{π}{3} = 0

= x

化简

\frac{5 π}{3} + 2 πn + \frac{π}{3} : 2 πn + 2 π

\frac{5 π}{3} + 2 πn + \frac{π}{3}

对同类项分组

= 2 πn + \frac{π}{3} + \frac{5 π}{3}

合并分式

\frac{π}{3} + \frac{5 π}{3} : 2 π

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + 5 π}{3}

同类项相加:

π + 5 π = 6 π

= \frac{6 π}{3}

数字相除:

\frac{6}{3} = 2

= 2 π

= 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

合并所有解

x = \frac{2 π}{3} + 2 πn, x = 2 πn + π, x = 2 πn + \frac{5 π}{3}, x = 2 πn + 2 π

作图

流行的例子

cos (θ) = - 0.89

cos (θ) = - 0.81

cos (θ) = - 0.44

cos (θ) = - 0.62

2-2sin^2(x)=2cos^2(x/2)

2 - 2 sin^{2} (x) = 2 cos^{2} (\frac{x}{2})