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受欢迎的三角函数 >

cos(4x)=sin(x),0<= x<= 2pi

解答

cos (4 x) = sin (x), 0 \leq x \leq 2 π

解答

x = \frac{π}{10}, x = \frac{π}{2}, x = \frac{9 π}{10}, x = \frac{13 π}{10}, x = \frac{17 π}{10}, x = \frac{7 π}{6}, x = \frac{11 π}{6}

+1

度数

x = 1 8^{\circ}, x = 9 0^{\circ}, x = 16 2^{\circ}, x = 23 4^{\circ}, x = 30 6^{\circ}, x = 21 0^{\circ}, x = 33 0^{\circ}

求解步骤

cos (4 x) = sin (x), 0 \leq x \leq 2 π

使用三角恒等式改写

cos (4 x) = sin (x)

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

cos (4 x) = sin (\frac{π}{2} - 4 x)

cos (4 x) = sin (\frac{π}{2} - 4 x)

使用反三角函数性质

cos (4 x) = sin (\frac{π}{2} - 4 x)

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

x = \frac{π}{2} - 4 x + 2 πn, x = π - (\frac{π}{2} - 4 x) + 2 πn

x = \frac{π}{2} - 4 x + 2 πn, x = π - (\frac{π}{2} - 4 x) + 2 πn

x = \frac{π}{2} - 4 x + 2 πn : x = \frac{π + 4 πn}{10}

x = \frac{π}{2} - 4 x + 2 πn

将

4 x

para o lado esquerdo

x = \frac{π}{2} - 4 x + 2 πn

两边加上

4 x

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

化简

5 x = \frac{π}{2} + 2 πn

5 x = \frac{π}{2} + 2 πn

两边除以

5

5 x = \frac{π}{2} + 2 πn

两边除以

5

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

化简

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

化简

\frac{5 x}{5} : x

\frac{5 x}{5}

数字相除:

\frac{5}{5} = 1

= x

化简

\frac{\frac{π}{2}}{5} + \frac{2 πn}{5} : \frac{π + 4 πn}{10}

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

使用法则

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

= \frac{\frac{π}{2} + 2 πn}{5}

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}}{5}

使用分式法则:

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

= \frac{π + 4 πn}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{π + 4 πn}{10}

x = \frac{π + 4 πn}{10}

x = \frac{π + 4 πn}{10}

x = \frac{π + 4 πn}{10}

x = π - (\frac{π}{2} - 4 x) + 2 πn : x = - \frac{π + 4 πn}{6}

x = π - (\frac{π}{2} - 4 x) + 2 πn

展开

π - (\frac{π}{2} - 4 x) + 2 πn : π - \frac{π}{2} + 4 x + 2 πn

π - (\frac{π}{2} - 4 x) + 2 πn

- (\frac{π}{2} - 4 x) : - \frac{π}{2} + 4 x

- (\frac{π}{2} - 4 x)

打开括号

= - (\frac{π}{2}) - (- 4 x)

使用加减运算法则

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

= - \frac{π}{2} + 4 x

= π - \frac{π}{2} + 4 x + 2 πn

x = π - \frac{π}{2} + 4 x + 2 πn

将

4 x

para o lado esquerdo

x = π - \frac{π}{2} + 4 x + 2 πn

两边减去

4 x

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

化简

- 3 x = π - \frac{π}{2} + 2 πn

- 3 x = π - \frac{π}{2} + 2 πn

两边除以

- 3

- 3 x = π - \frac{π}{2} + 2 πn

两边除以

- 3

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

化简

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

化简

\frac{- 3 x}{- 3} : x

\frac{- 3 x}{- 3}

使用分式法则:

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

= \frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

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

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

使用分式法则:

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

= - \frac{π - \frac{π}{2} + 2 πn}{3}

化简

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

π - \frac{π}{2} + 2 πn

将项转换为分式:

π = \frac{π 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 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 - π + 2 πn \cdot 2}{2}

π 2 - π + 2 πn \cdot 2 = π + 4 πn

π 2 - π + 2 πn \cdot 2

同类项相加:

2 π - π = π

= π + 2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= π + 4 πn

= \frac{π + 4 πn}{2}

= - \frac{\frac{π + 4 πn}{2}}{3}

化简

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

\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}

= - \frac{π + 4 πn}{6}

x = - \frac{π + 4 πn}{6}

x = - \frac{π + 4 πn}{6}

x = - \frac{π + 4 πn}{6}

在

0 \leq x \leq 2 π

范围内的解

x = \frac{π}{10}, x = \frac{π}{2}, x = \frac{9 π}{10}, x = \frac{13 π}{10}, x = \frac{17 π}{10}, x = \frac{7 π}{6}, x = \frac{11 π}{6}

作图

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