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受欢迎的三角函数 >

sqrt(1-cos(3x))=2pi

解答

1 - cos (3 x) = 2 π

解答

x \in R 无解

求解步骤

1 - cos (3 x) = 2 π

两边进行平方:

1 - cos (3 x) = 4 π^{2}

1 - cos (3 x) = 2 π

(1 - cos (3 x))^{2} = (2 π)^{2}

展开

(1 - cos (3 x))^{2} : 1 - cos (3 x)

(1 - cos (3 x))^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= ((1 - cos (3 x))^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= (1 - cos (3 x))^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 1 - cos (3 x)

展开

(2 π)^{2} : 4 π^{2}

(2 π)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} π^{2}

2^{2} = 4

= 4 π^{2}

1 - cos (3 x) = 4 π^{2}

1 - cos (3 x) = 4 π^{2}

解

1 - cos (3 x) = 4 π^{2} : cos (3 x) = - 4 π^{2} + 1

1 - cos (3 x) = 4 π^{2}

将

1

到右边

1 - cos (3 x) = 4 π^{2}

两边减去

1

1 - cos (3 x) - 1 = 4 π^{2} - 1

化简

- cos (3 x) = 4 π^{2} - 1

- cos (3 x) = 4 π^{2} - 1

两边除以

- 1

- cos (3 x) = 4 π^{2} - 1

两边除以

- 1

\frac{- cos ( 3 x )}{- 1} = \frac{4 π ^{2}}{- 1} - \frac{1}{- 1}

化简

\frac{- cos ( 3 x )}{- 1} = \frac{4 π ^{2}}{- 1} - \frac{1}{- 1}

化简

\frac{- cos ( 3 x )}{- 1} : cos (3 x)

\frac{- cos ( 3 x )}{- 1}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{cos ( 3 x )}{1}

使用法则

\frac{a}{1} = a

= cos (3 x)

化简

\frac{4 π ^{2}}{- 1} - \frac{1}{- 1} : - 4 π^{2} + 1

\frac{4 π ^{2}}{- 1} - \frac{1}{- 1}

使用法则

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

= \frac{4 π ^{2} - 1}{- 1}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{4 π ^{2} - 1}{1}

使用法则

\frac{a}{1} = a

= - (4 π^{2} - 1)

打开括号

= - (4 π^{2}) - (- 1)

使用加减运算法则

- (- a) = a, - (a) = - a

= - 4 π^{2} + 1

cos (3 x) = - 4 π^{2} + 1

cos (3 x) = - 4 π^{2} + 1

cos (3 x) = - 4 π^{2} + 1

cos (3 x) = - 4 π^{2} + 1

验证解:

cos (3 x) = - 4 π^{2} + 1

真

将它们代入

1 - cos (3 x) = 2 π

检验解是否符合

去除与方程不符的解。

代入

cos (3 x) = - 4 π^{2} + 1 :

真

1 - (- 4 π^{2} + 1) = 2 π

1 - (- 4 π^{2} + 1) = 2 π

1 - (- 4 π^{2} + 1)

乘开

1 - (- 4 π^{2} + 1) : 4 π^{2}

1 - (- 4 π^{2} + 1)

- (- 4 π^{2} + 1) : 4 π^{2} - 1

- (- 4 π^{2} + 1)

打开括号

= - (- 4 π^{2}) - (1)

使用加减运算法则

- (- a) = a, - (a) = - a

= 4 π^{2} - 1

= 1 + 4 π^{2} - 1

1 - 1 = 0

= 4 π^{2}

= 4 π^{2}

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= 4 π^{2}

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

= 2 π^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

π^{2} = π

= 2 π

2 π = 2 π

真

解是

cos (3 x) = - 4 π^{2} + 1

- 1 \leq cos (x) \leq 1

x \in R 无解

作图

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