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sin(5x+8)=cos(9x-16)

解答

sin (5 x + 8) = cos (9 x - 16)

解答

x = \frac{4 πn + 16 + π}{28}, x = - \frac{π + 4 πn - 48}{8}

+1

度数

x = 39.16901 \dots^{\circ} + 25.71428 \dots^{\circ} n, x = 321.27467 \dots^{\circ} - 9 0^{\circ} n

求解步骤

sin (5 x + 8) = cos (9 x - 16)

使用三角恒等式改写

sin (5 x + 8) = cos (9 x - 16)

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x + 8) = sin (\frac{π}{2} - (9 x - 16))

sin (5 x + 8) = sin (\frac{π}{2} - (9 x - 16))

使用反三角函数性质

sin (5 x + 8) = sin (\frac{π}{2} - (9 x - 16))

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

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn, 5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn, 5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn : x = \frac{4 πn + 16 + π}{28}

5 x + 8 = \frac{π}{2} - (9 x - 16) + 2 πn

展开

\frac{π}{2} - (9 x - 16) + 2 πn : \frac{π}{2} - 9 x + 16 + 2 πn

\frac{π}{2} - (9 x - 16) + 2 πn

- (9 x - 16) : - 9 x + 16

- (9 x - 16)

打开括号

= - (9 x) - (- 16)

使用加减运算法则

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

= - 9 x + 16

= \frac{π}{2} - 9 x + 16 + 2 πn

5 x + 8 = \frac{π}{2} - 9 x + 16 + 2 πn

将

8

到右边

5 x + 8 = \frac{π}{2} - 9 x + 16 + 2 πn

两边减去

8

5 x + 8 - 8 = \frac{π}{2} - 9 x + 16 + 2 πn - 8

化简

5 x + 8 - 8 = \frac{π}{2} - 9 x + 16 + 2 πn - 8

化简

5 x + 8 - 8 : 5 x

5 x + 8 - 8

同类项相加:

8 - 8 = 0

= 5 x

化简

\frac{π}{2} - 9 x + 16 + 2 πn - 8 : - 9 x + 2 πn + 8 + \frac{π}{2}

\frac{π}{2} - 9 x + 16 + 2 πn - 8

对同类项分组

= - 9 x + 2 πn + \frac{π}{2} + 16 - 8

数字相加/相减:

16 - 8 = 8

= - 9 x + 2 πn + 8 + \frac{π}{2}

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

将

9 x

para o lado esquerdo

5 x = - 9 x + 2 πn + 8 + \frac{π}{2}

两边加上

9 x

5 x + 9 x = - 9 x + 2 πn + 8 + \frac{π}{2} + 9 x

化简

14 x = 2 πn + 8 + \frac{π}{2}

14 x = 2 πn + 8 + \frac{π}{2}

两边除以

14

14 x = 2 πn + 8 + \frac{π}{2}

两边除以

14

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{8}{14} + \frac{\frac{π}{2}}{14}

化简

\frac{14 x}{14} = \frac{2 πn}{14} + \frac{8}{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{8}{14} + \frac{\frac{π}{2}}{14} : \frac{4 πn + 16 + π}{28}

\frac{2 πn}{14} + \frac{8}{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 + 8 + \frac{π}{2}}{14}

化简

2 πn + 8 + \frac{π}{2} : \frac{4 πn + 16 + π}{2}

2 πn + 8 + \frac{π}{2}

将项转换为分式:

2 πn = \frac{2 πn 2}{2}, 8 = \frac{8 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{8 \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 πn \cdot 2 + 8 \cdot 2 + π}{2}

2 πn \cdot 2 + 8 \cdot 2 + π = 4 πn + 16 + π

2 πn \cdot 2 + 8 \cdot 2 + π

数字相乘:

2 \cdot 2 = 4

= 4 πn + 8 \cdot 2 + π

数字相乘:

8 \cdot 2 = 16

= 4 πn + 16 + π

= \frac{4 πn + 16 + π}{2}

= \frac{\frac{4 πn + 16 + π}{2}}{14}

使用分式法则:

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

= \frac{4 πn + 16 + π}{2 \cdot 14}

数字相乘:

2 \cdot 14 = 28

= \frac{4 πn + 16 + π}{28}

x = \frac{4 πn + 16 + π}{28}

x = \frac{4 πn + 16 + π}{28}

x = \frac{4 πn + 16 + π}{28}

5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn : x = - \frac{π + 4 πn - 48}{8}

5 x + 8 = π - (\frac{π}{2} - (9 x - 16)) + 2 πn

展开

π - (\frac{π}{2} - (9 x - 16)) + 2 πn : π - \frac{π}{2} + 9 x - 16 + 2 πn

π - (\frac{π}{2} - (9 x - 16)) + 2 πn

- (9 x - 16) : - 9 x + 16

- (9 x - 16)

打开括号

= - (9 x) - (- 16)

使用加减运算法则

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

= - 9 x + 16

= π - (- 9 x + 16 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 9 x + 16) : - \frac{π}{2} + 9 x - 16

- (\frac{π}{2} - 9 x + 16)

打开括号

= - (\frac{π}{2}) - (- 9 x) - (16)

使用加减运算法则

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

= - \frac{π}{2} + 9 x - 16

= π - \frac{π}{2} + 9 x - 16 + 2 πn

5 x + 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn

将

8

到右边

5 x + 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn

两边减去

8

5 x + 8 - 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn - 8

化简

5 x + 8 - 8 = π - \frac{π}{2} + 9 x - 16 + 2 πn - 8

化简

5 x + 8 - 8 : 5 x

5 x + 8 - 8

同类项相加:

8 - 8 = 0

= 5 x

化简

π - \frac{π}{2} + 9 x - 16 + 2 πn - 8 : 9 x + 2 πn + π - 24 - \frac{π}{2}

π - \frac{π}{2} + 9 x - 16 + 2 πn - 8

对同类项分组

= 9 x + π + 2 πn - \frac{π}{2} - 16 - 8

数字相减:

- 16 - 8 = - 24

= 9 x + 2 πn + π - 24 - \frac{π}{2}

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

将

9 x

para o lado esquerdo

5 x = 9 x + 2 πn + π - 24 - \frac{π}{2}

两边减去

9 x

5 x - 9 x = 9 x + 2 πn + π - 24 - \frac{π}{2} - 9 x

化简

- 4 x = 2 πn + π - 24 - \frac{π}{2}

- 4 x = 2 πn + π - 24 - \frac{π}{2}

两边除以

- 4

- 4 x = 2 πn + π - 24 - \frac{π}{2}

两边除以

- 4

\frac{- 4 x}{- 4} = \frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 4}

化简

\frac{- 4 x}{- 4} = \frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 4}

化简

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

使用分式法则:

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

= \frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 4} : - \frac{π + 4 πn - 48}{8}

\frac{2 πn}{- 4} + \frac{π}{- 4} - \frac{24}{- 4} - \frac{\frac{π}{2}}{- 4}

使用法则

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

= \frac{2 πn + π - 24 - \frac{π}{2}}{- 4}

使用分式法则:

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

= - \frac{2 πn + π - 24 - \frac{π}{2}}{4}

化简

2 πn + π - 24 - \frac{π}{2} : \frac{π + 4 πn - 48}{2}

2 πn + π - 24 - \frac{π}{2}

将项转换为分式:

2 πn = \frac{2 πn 2}{2}, π = \frac{π 2}{2}, 24 = \frac{24 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π 2}{2} - \frac{24 \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 πn \cdot 2 + π 2 - 24 \cdot 2 - π}{2}

2 πn \cdot 2 + π 2 - 24 \cdot 2 - π = π + 4 πn - 48

2 πn \cdot 2 + π 2 - 24 \cdot 2 - π

对同类项分组

= 2 π - π + 2 \cdot 2 πn - 24 \cdot 2

同类项相加:

2 π - π = π

= π + 2 \cdot 2 πn - 24 \cdot 2

数字相乘:

2 \cdot 2 = 4

= π + 4 πn - 24 \cdot 2

数字相乘:

24 \cdot 2 = 48

= π + 4 πn - 48

= \frac{π + 4 πn - 48}{2}

= - \frac{\frac{π + 4 πn - 48}{2}}{4}

化简

\frac{\frac{π + 4 πn - 48}{2}}{4} : \frac{π + 4 πn - 48}{8}

\frac{\frac{π + 4 πn - 48}{2}}{4}

使用分式法则:

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

= \frac{π + 4 πn - 48}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{π + 4 πn - 48}{8}

= - \frac{π + 4 πn - 48}{8}

x = - \frac{π + 4 πn - 48}{8}

x = - \frac{π + 4 πn - 48}{8}

x = - \frac{π + 4 πn - 48}{8}

x = \frac{4 πn + 16 + π}{28}, x = - \frac{π + 4 πn - 48}{8}

x = \frac{4 πn + 16 + π}{28}, x = - \frac{π + 4 πn - 48}{8}

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cos (x^{\circ}) = \frac{125}{174}

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csc (θ) = 23

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60 sin (4 5^{\circ}) = 400 sin (θ)

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