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受欢迎的三角函数 >

cos(7x)=sin(4x+2)

解答

cos (7 x) = sin (4 x + 2)

解答

x = \frac{- 4 + 4 πn + π}{22}, x = - \frac{π + 4 πn - 4}{6}

+1

度数

x = - 2.23559 \dots^{\circ} + 32.72727 \dots^{\circ} n, x = 8.19718 \dots^{\circ} - 12 0^{\circ} n

求解步骤

cos (7 x) = sin (4 x + 2)

使用三角恒等式改写

cos (7 x) = sin (4 x + 2)

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

cos (7 x) = sin (\frac{π}{2} - 7 x)

cos (7 x) = sin (\frac{π}{2} - 7 x)

使用反三角函数性质

cos (7 x) = sin (\frac{π}{2} - 7 x)

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

4 x + 2 = \frac{π}{2} - 7 x + 2 πn, 4 x + 2 = π - (\frac{π}{2} - 7 x) + 2 πn

4 x + 2 = \frac{π}{2} - 7 x + 2 πn, 4 x + 2 = π - (\frac{π}{2} - 7 x) + 2 πn

4 x + 2 = \frac{π}{2} - 7 x + 2 πn : x = \frac{- 4 + 4 πn + π}{22}

4 x + 2 = \frac{π}{2} - 7 x + 2 πn

将

2

到右边

4 x + 2 = \frac{π}{2} - 7 x + 2 πn

两边减去

2

4 x + 2 - 2 = \frac{π}{2} - 7 x + 2 πn - 2

化简

4 x = \frac{π}{2} - 7 x + 2 πn - 2

4 x = \frac{π}{2} - 7 x + 2 πn - 2

将

7 x

para o lado esquerdo

4 x = \frac{π}{2} - 7 x + 2 πn - 2

两边加上

7 x

4 x + 7 x = \frac{π}{2} - 7 x + 2 πn - 2 + 7 x

化简

11 x = \frac{π}{2} + 2 πn - 2

11 x = \frac{π}{2} + 2 πn - 2

两边除以

11

11 x = \frac{π}{2} + 2 πn - 2

两边除以

11

\frac{11 x}{11} = \frac{\frac{π}{2}}{11} + \frac{2 πn}{11} - \frac{2}{11}

化简

\frac{11 x}{11} = \frac{\frac{π}{2}}{11} + \frac{2 πn}{11} - \frac{2}{11}

化简

\frac{11 x}{11} : x

\frac{11 x}{11}

数字相除:

\frac{11}{11} = 1

= x

化简

\frac{\frac{π}{2}}{11} + \frac{2 πn}{11} - \frac{2}{11} : \frac{- 4 + 4 πn + π}{22}

\frac{\frac{π}{2}}{11} + \frac{2 πn}{11} - \frac{2}{11}

对同类项分组

= - \frac{2}{11} + \frac{2 πn}{11} + \frac{\frac{π}{2}}{11}

使用法则

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

= \frac{- 2 + 2 πn + \frac{π}{2}}{11}

化简

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

- 2 + 2 πn + \frac{π}{2}

将项转换为分式:

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

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 4 + 4 πn + π}{2}

= \frac{\frac{- 4 + 4 πn + π}{2}}{11}

使用分式法则:

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

= \frac{- 4 + 4 πn + π}{2 \cdot 11}

数字相乘:

2 \cdot 11 = 22

= \frac{- 4 + 4 πn + π}{22}

x = \frac{- 4 + 4 πn + π}{22}

x = \frac{- 4 + 4 πn + π}{22}

x = \frac{- 4 + 4 πn + π}{22}

4 x + 2 = π - (\frac{π}{2} - 7 x) + 2 πn : x = - \frac{π + 4 πn - 4}{6}

4 x + 2 = π - (\frac{π}{2} - 7 x) + 2 πn

展开

π - (\frac{π}{2} - 7 x) + 2 πn : π - \frac{π}{2} + 7 x + 2 πn

π - (\frac{π}{2} - 7 x) + 2 πn

- (\frac{π}{2} - 7 x) : - \frac{π}{2} + 7 x

- (\frac{π}{2} - 7 x)

打开括号

= - (\frac{π}{2}) - (- 7 x)

使用加减运算法则

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

= - \frac{π}{2} + 7 x

= π - \frac{π}{2} + 7 x + 2 πn

4 x + 2 = π - \frac{π}{2} + 7 x + 2 πn

将

2

到右边

4 x + 2 = π - \frac{π}{2} + 7 x + 2 πn

两边减去

2

4 x + 2 - 2 = π - \frac{π}{2} + 7 x + 2 πn - 2

化简

4 x = π - \frac{π}{2} + 7 x + 2 πn - 2

4 x = π - \frac{π}{2} + 7 x + 2 πn - 2

将

7 x

para o lado esquerdo

4 x = π - \frac{π}{2} + 7 x + 2 πn - 2

两边减去

7 x

4 x - 7 x = π - \frac{π}{2} + 7 x + 2 πn - 2 - 7 x

化简

- 3 x = π - \frac{π}{2} + 2 πn - 2

- 3 x = π - \frac{π}{2} + 2 πn - 2

两边除以

- 3

- 3 x = π - \frac{π}{2} + 2 πn - 2

两边除以

- 3

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

化简

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

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

对同类项分组

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

使用法则

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

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

使用分式法则:

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

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

化简

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

π - 2 + 2 πn - \frac{π}{2}

将项转换为分式:

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

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

π 2 - 2 \cdot 2 + 2 πn \cdot 2 - π = π + 4 πn - 4

π 2 - 2 \cdot 2 + 2 πn \cdot 2 - π

对同类项分组

= 2 π - π + 2 \cdot 2 πn - 2 \cdot 2

同类项相加:

2 π - π = π

= π + 2 \cdot 2 πn - 2 \cdot 2

数字相乘:

2 \cdot 2 = 4

= π + 4 πn - 4

= \frac{π + 4 πn - 4}{2}

= - \frac{\frac{π + 4 πn - 4}{2}}{3}

化简

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

\frac{\frac{π + 4 πn - 4}{2}}{3}

使用分式法则:

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

= \frac{π + 4 πn - 4}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{π + 4 πn - 4}{6}

= - \frac{π + 4 πn - 4}{6}

x = - \frac{π + 4 πn - 4}{6}

x = - \frac{π + 4 πn - 4}{6}

x = - \frac{π + 4 πn - 4}{6}

x = \frac{- 4 + 4 πn + π}{22}, x = - \frac{π + 4 πn - 4}{6}

x = \frac{- 4 + 4 πn + π}{22}, x = - \frac{π + 4 πn - 4}{6}

作图

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arctan(θ)= 6/8

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