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受欢迎的三角函数 >

sin(9x+40)=cos(-5x+42)

解答

sin (9 x + 40) = cos (- 5 x + 42)

解答

x = \frac{4 πn + π - 164}{8}, x = \frac{4 πn + 4 + π}{28}

+1

度数

x = - 1152.06348 \dots^{\circ} + 9 0^{\circ} n, x = 14.61368 \dots^{\circ} + 25.71428 \dots^{\circ} n

求解步骤

sin (9 x + 40) = cos (- 5 x + 42)

使用三角恒等式改写

sin (9 x + 40) = cos (- 5 x + 42)

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

sin (9 x + 40) = sin (\frac{π}{2} - (- 5 x + 42))

sin (9 x + 40) = sin (\frac{π}{2} - (- 5 x + 42))

使用反三角函数性质

sin (9 x + 40) = sin (\frac{π}{2} - (- 5 x + 42))

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

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn, 9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn, 9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn : x = \frac{4 πn + π - 164}{8}

9 x + 40 = \frac{π}{2} - (- 5 x + 42) + 2 πn

展开

\frac{π}{2} - (- 5 x + 42) + 2 πn : \frac{π}{2} + 5 x - 42 + 2 πn

\frac{π}{2} - (- 5 x + 42) + 2 πn

- (- 5 x + 42) : 5 x - 42

- (- 5 x + 42)

打开括号

= - (- 5 x) - (42)

使用加减运算法则

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

= 5 x - 42

= \frac{π}{2} + 5 x - 42 + 2 πn

9 x + 40 = \frac{π}{2} + 5 x - 42 + 2 πn

将

40

到右边

9 x + 40 = \frac{π}{2} + 5 x - 42 + 2 πn

两边减去

40

9 x + 40 - 40 = \frac{π}{2} + 5 x - 42 + 2 πn - 40

化简

9 x + 40 - 40 = \frac{π}{2} + 5 x - 42 + 2 πn - 40

化简

9 x + 40 - 40 : 9 x

9 x + 40 - 40

同类项相加:

40 - 40 = 0

= 9 x

化简

\frac{π}{2} + 5 x - 42 + 2 πn - 40 : 5 x + 2 πn + \frac{π}{2} - 82

\frac{π}{2} + 5 x - 42 + 2 πn - 40

对同类项分组

= 5 x + 2 πn + \frac{π}{2} - 42 - 40

数字相减:

- 42 - 40 = - 82

= 5 x + 2 πn + \frac{π}{2} - 82

9 x = 5 x + 2 πn + \frac{π}{2} - 82

9 x = 5 x + 2 πn + \frac{π}{2} - 82

9 x = 5 x + 2 πn + \frac{π}{2} - 82

将

5 x

para o lado esquerdo

9 x = 5 x + 2 πn + \frac{π}{2} - 82

两边减去

5 x

9 x - 5 x = 5 x + 2 πn + \frac{π}{2} - 82 - 5 x

化简

4 x = 2 πn + \frac{π}{2} - 82

4 x = 2 πn + \frac{π}{2} - 82

两边除以

4

4 x = 2 πn + \frac{π}{2} - 82

两边除以

4

\frac{4 x}{4} = \frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4}

化简

\frac{4 x}{4} = \frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4}

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4} : \frac{4 πn + π - 164}{8}

\frac{2 πn}{4} + \frac{\frac{π}{2}}{4} - \frac{82}{4}

使用法则

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

= \frac{2 πn + \frac{π}{2} - 82}{4}

化简

2 πn + \frac{π}{2} - 82 : \frac{4 πn + π - 164}{2}

2 πn + \frac{π}{2} - 82

将项转换为分式:

2 πn = \frac{2 πn 2}{2}, 82 = \frac{82 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{82 \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{2 πn \cdot 2 + π - 82 \cdot 2}{2}

2 πn \cdot 2 + π - 82 \cdot 2 = 4 πn + π - 164

2 πn \cdot 2 + π - 82 \cdot 2

数字相乘:

2 \cdot 2 = 4

= 4 πn + π - 82 \cdot 2

数字相乘:

82 \cdot 2 = 164

= 4 πn + π - 164

= \frac{4 πn + π - 164}{2}

= \frac{\frac{4 πn + π - 164}{2}}{4}

使用分式法则:

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

= \frac{4 πn + π - 164}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{4 πn + π - 164}{8}

x = \frac{4 πn + π - 164}{8}

x = \frac{4 πn + π - 164}{8}

x = \frac{4 πn + π - 164}{8}

9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn : x = \frac{4 πn + 4 + π}{28}

9 x + 40 = π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

展开

π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn : π - \frac{π}{2} - 5 x + 42 + 2 πn

π - (\frac{π}{2} - (- 5 x + 42)) + 2 πn

- (- 5 x + 42) : 5 x - 42

- (- 5 x + 42)

打开括号

= - (- 5 x) - (42)

使用加减运算法则

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

= 5 x - 42

= π - (5 x + \frac{π}{2} - 42) + 2 πn

- (\frac{π}{2} + 5 x - 42) : - \frac{π}{2} - 5 x + 42

- (\frac{π}{2} + 5 x - 42)

打开括号

= - (\frac{π}{2}) - (5 x) - (- 42)

使用加减运算法则

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

= - \frac{π}{2} - 5 x + 42

= π - \frac{π}{2} - 5 x + 42 + 2 πn

9 x + 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn

将

40

到右边

9 x + 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn

两边减去

40

9 x + 40 - 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn - 40

化简

9 x + 40 - 40 = π - \frac{π}{2} - 5 x + 42 + 2 πn - 40

化简

9 x + 40 - 40 : 9 x

9 x + 40 - 40

同类项相加:

40 - 40 = 0

= 9 x

化简

π - \frac{π}{2} - 5 x + 42 + 2 πn - 40 : - 5 x + 2 πn + 2 + π - \frac{π}{2}

π - \frac{π}{2} - 5 x + 42 + 2 πn - 40

对同类项分组

= - 5 x + π + 2 πn - \frac{π}{2} + 42 - 40

数字相加/相减:

42 - 40 = 2

= - 5 x + 2 πn + 2 + π - \frac{π}{2}

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

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

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

将

5 x

para o lado esquerdo

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

两边加上

5 x

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

化简

14 x = 2 πn + 2 + π - \frac{π}{2}

14 x = 2 πn + 2 + π - \frac{π}{2}

两边除以

14

14 x = 2 πn + 2 + π - \frac{π}{2}

两边除以

14

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{2}{14} + \frac{π}{14} - \frac{\frac{π}{2}}{14}

化简

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{2}{14} + \frac{π}{14} - \frac{\frac{π}{2}}{14}

化简

\frac{14 x}{14} : x

\frac{14 x}{14}

数字相除:

\frac{14}{14} = 1

= x

化简

\frac{2 πn}{14} + \frac{2}{14} + \frac{π}{14} - \frac{\frac{π}{2}}{14} : \frac{4 πn + 4 + π}{28}

\frac{2 πn}{14} + \frac{2}{14} + \frac{π}{14} - \frac{\frac{π}{2}}{14}

使用法则

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

= \frac{2 πn + 2 + π - \frac{π}{2}}{14}

化简

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

2 πn + 2 + π - \frac{π}{2}

将项转换为分式:

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

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

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

2 πn \cdot 2 + 2 \cdot 2 + π 2 - π

同类项相加:

2 π - π = π

= 2 \cdot 2 πn + 2 \cdot 2 + π

数字相乘:

2 \cdot 2 = 4

= 4 πn + 4 + π

= \frac{4 πn + 4 + π}{2}

= \frac{\frac{4 πn + 4 + π}{2}}{14}

使用分式法则:

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

= \frac{4 πn + 4 + π}{2 \cdot 14}

数字相乘:

2 \cdot 14 = 28

= \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + π - 164}{8}, x = \frac{4 πn + 4 + π}{28}

x = \frac{4 πn + π - 164}{8}, x = \frac{4 πn + 4 + π}{28}

作图

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