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cos^2(3x)=cos(9x^2)

解答

cos^{2} (3 x) = cos (9 x^{2})

解答

x = \frac{π + 4 πn}{6}, x = \frac{3 π + 4 πn}{6}, x = \frac{2 πn}{3}

+1

度数

x = 3 0^{\circ} + 12 0^{\circ} n, x = 9 0^{\circ} + 12 0^{\circ} n, x = 0^{\circ} + 12 0^{\circ} n

求解步骤

cos^{2} (3 x) = cos (9 x^{2})

两边减去

cos (9 x^{2})

cos^{2} (3 x) - cos (9 x^{2}) = 0

令

: u = 3 x

cos^{2} (u) - cos (\frac{u ^{2}}{u}) = 0

分解

cos^{2} (u) - cos (\frac{u ^{2}}{u}) : cos (u) (cos (u) - 1)

cos^{2} (u) - cos (\frac{u ^{2}}{u})

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

\frac{u ^{2}}{u}

约分:

u

= u

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

使用指数法则:

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

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

= cos (u) cos (u) - cos (u)

因式分解出通项

cos (u)

= cos (u) (cos (u) - 1)

cos (u) (cos (u) - 1) = 0

分别求解每个部分

cos (u) = 0 or cos (u) - 1 = 0

cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

cos (u) = 0

cos (u) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

cos (u) - 1 = 0 : u = 2 πn

cos (u) - 1 = 0

将

1

到右边

cos (u) - 1 = 0

两边加上

1

cos (u) - 1 + 1 = 0 + 1

化简

cos (u) = 1

cos (u) = 1

cos (u) = 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = 0 + 2 πn

u = 0 + 2 πn

解

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn

合并所有解

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

u = 3 x

代回

3 x = \frac{π}{2} + 2 πn : x = \frac{π + 4 πn}{6}

3 x = \frac{π}{2} + 2 πn

两边除以

3

3 x = \frac{π}{2} + 2 πn

两边除以

3

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

化简

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

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} : \frac{π + 4 πn}{6}

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

使用法则

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

= \frac{\frac{π}{2} + 2 πn}{3}

化简

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

\frac{π}{2} + 2 πn

将项转换为分式:

2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} + \frac{2 πn \cdot 2}{2}

因为分母相等，所以合并分式:

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

= \frac{π + 2 πn \cdot 2}{2}

数字相乘:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{3}

使用分式法则:

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

= \frac{π + 4 πn}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{π + 4 πn}{6}

x = \frac{π + 4 πn}{6}

x = \frac{π + 4 πn}{6}

x = \frac{π + 4 πn}{6}

3 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π + 4 πn}{6}

3 x = \frac{3 π}{2} + 2 πn

两边除以

3

3 x = \frac{3 π}{2} + 2 πn

两边除以

3

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

化简

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

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3} : \frac{3 π + 4 πn}{6}

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

使用法则

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

= \frac{\frac{3 π}{2} + 2 πn}{3}

化简

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

\frac{3 π}{2} + 2 πn

将项转换为分式:

2 πn = \frac{2 πn 2}{2}

= \frac{3 π}{2} + \frac{2 πn \cdot 2}{2}

因为分母相等，所以合并分式:

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

= \frac{3 π + 2 πn \cdot 2}{2}

数字相乘:

2 \cdot 2 = 4

= \frac{3 π + 4 πn}{2}

= \frac{\frac{3 π + 4 πn}{2}}{3}

使用分式法则:

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

= \frac{3 π + 4 πn}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{3 π + 4 πn}{6}

x = \frac{3 π + 4 πn}{6}

x = \frac{3 π + 4 πn}{6}

x = \frac{3 π + 4 πn}{6}

3 x = 2 πn : x = \frac{2 πn}{3}

3 x = 2 πn

两边除以

3

3 x = 2 πn

两边除以

3

\frac{3 x}{3} = \frac{2 πn}{3}

化简

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

x = \frac{π + 4 πn}{6}, x = \frac{3 π + 4 πn}{6}, x = \frac{2 πn}{3}

因为方程对以下值无定义:

\frac{π + 4 πn}{6}, \frac{3 π + 4 πn}{6}, \frac{2 πn}{3}

x = \frac{π + 4 πn}{6}, x = \frac{3 π + 4 πn}{6}, x = \frac{2 πn}{3}

流行的例子

sqrt(2)cos(θ)-sqrt(2)=0

2 cos (θ) - 2 = 0

solvefor x,y=sqrt(r(sec(x)))求解

x, y = r (sec (x))

csc^2(x)-3=6tan(x)

csc^{2} (x) - 3 = 6 tan (x)

solvefor C,arctan((1/(2*pi*R*C))/f)=0 求解

C, arctan (\frac{\frac{1}{2 \cdot π \cdot R \cdot C}}{f}) = 0

sin(x)=-2cos(x)

sin (x) = - 2 cos (x)