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受欢迎的三角函数 >

cos(2x-1)=sin(3x+6)

解答

cos (2 x - 1) = sin (3 x + 6)

解答

x = \frac{4 πn + π - 10}{10}, x = 2 πn + π - 7 - \frac{π}{2}

+1

度数

x = - 39.29577 \dots^{\circ} + 7 2^{\circ} n, x = - 311.07045 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos (2 x - 1) = sin (3 x + 6)

使用三角恒等式改写

cos (2 x - 1) = sin (3 x + 6)

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

cos (2 x - 1) = sin (\frac{π}{2} - (2 x - 1))

cos (2 x - 1) = sin (\frac{π}{2} - (2 x - 1))

使用反三角函数性质

cos (2 x - 1) = sin (\frac{π}{2} - (2 x - 1))

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

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn, 3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn, 3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn : x = \frac{4 πn + π - 10}{10}

3 x + 6 = \frac{π}{2} - (2 x - 1) + 2 πn

展开

\frac{π}{2} - (2 x - 1) + 2 πn : \frac{π}{2} - 2 x + 1 + 2 πn

\frac{π}{2} - (2 x - 1) + 2 πn

- (2 x - 1) : - 2 x + 1

- (2 x - 1)

打开括号

= - (2 x) - (- 1)

使用加减运算法则

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

= - 2 x + 1

= \frac{π}{2} - 2 x + 1 + 2 πn

3 x + 6 = \frac{π}{2} - 2 x + 1 + 2 πn

将

6

到右边

3 x + 6 = \frac{π}{2} - 2 x + 1 + 2 πn

两边减去

6

3 x + 6 - 6 = \frac{π}{2} - 2 x + 1 + 2 πn - 6

化简

3 x + 6 - 6 = \frac{π}{2} - 2 x + 1 + 2 πn - 6

化简

3 x + 6 - 6 : 3 x

3 x + 6 - 6

同类项相加:

6 - 6 = 0

= 3 x

化简

\frac{π}{2} - 2 x + 1 + 2 πn - 6 : - 2 x + 2 πn + \frac{π}{2} - 5

\frac{π}{2} - 2 x + 1 + 2 πn - 6

对同类项分组

= - 2 x + 2 πn + \frac{π}{2} + 1 - 6

数字相加/相减:

1 - 6 = - 5

= - 2 x + 2 πn + \frac{π}{2} - 5

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

将

2 x

para o lado esquerdo

3 x = - 2 x + 2 πn + \frac{π}{2} - 5

两边加上

2 x

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

化简

5 x = 2 πn + \frac{π}{2} - 5

5 x = 2 πn + \frac{π}{2} - 5

两边除以

5

5 x = 2 πn + \frac{π}{2} - 5

两边除以

5

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

化简

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

化简

\frac{5 x}{5} : x

\frac{5 x}{5}

数字相除:

\frac{5}{5} = 1

= x

化简

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

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

使用法则

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

= \frac{2 πn + \frac{π}{2} - 5}{5}

化简

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

2 πn + \frac{π}{2} - 5

将项转换为分式:

2 πn = \frac{2 πn 2}{2}, 5 = \frac{5 \cdot 2}{2}

= \frac{2 πn \cdot 2}{2} + \frac{π}{2} - \frac{5 \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 - 5 \cdot 2}{2}

2 πn \cdot 2 + π - 5 \cdot 2 = 4 πn + π - 10

2 πn \cdot 2 + π - 5 \cdot 2

数字相乘:

2 \cdot 2 = 4

= 4 πn + π - 5 \cdot 2

数字相乘:

5 \cdot 2 = 10

= 4 πn + π - 10

= \frac{4 πn + π - 10}{2}

= \frac{\frac{4 πn + π - 10}{2}}{5}

使用分式法则:

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

= \frac{4 πn + π - 10}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{4 πn + π - 10}{10}

x = \frac{4 πn + π - 10}{10}

x = \frac{4 πn + π - 10}{10}

x = \frac{4 πn + π - 10}{10}

3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn : x = 2 πn + π - 7 - \frac{π}{2}

3 x + 6 = π - (\frac{π}{2} - (2 x - 1)) + 2 πn

展开

π - (\frac{π}{2} - (2 x - 1)) + 2 πn : π - \frac{π}{2} + 2 x - 1 + 2 πn

π - (\frac{π}{2} - (2 x - 1)) + 2 πn

- (2 x - 1) : - 2 x + 1

- (2 x - 1)

打开括号

= - (2 x) - (- 1)

使用加减运算法则

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

= - 2 x + 1

= π - (- 2 x + 1 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 2 x + 1) : - \frac{π}{2} + 2 x - 1

- (\frac{π}{2} - 2 x + 1)

打开括号

= - (\frac{π}{2}) - (- 2 x) - (1)

使用加减运算法则

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

= - \frac{π}{2} + 2 x - 1

= π - \frac{π}{2} + 2 x - 1 + 2 πn

3 x + 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn

将

6

到右边

3 x + 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn

两边减去

6

3 x + 6 - 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn - 6

化简

3 x + 6 - 6 = π - \frac{π}{2} + 2 x - 1 + 2 πn - 6

化简

3 x + 6 - 6 : 3 x

3 x + 6 - 6

同类项相加:

6 - 6 = 0

= 3 x

化简

π - \frac{π}{2} + 2 x - 1 + 2 πn - 6 : 2 x + 2 πn + π - 7 - \frac{π}{2}

π - \frac{π}{2} + 2 x - 1 + 2 πn - 6

对同类项分组

= 2 x + π + 2 πn - \frac{π}{2} - 1 - 6

数字相减:

- 1 - 6 = - 7

= 2 x + 2 πn + π - 7 - \frac{π}{2}

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

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

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

将

2 x

para o lado esquerdo

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

两边减去

2 x

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

化简

x = 2 πn + π - 7 - \frac{π}{2}

x = 2 πn + π - 7 - \frac{π}{2}

x = \frac{4 πn + π - 10}{10}, x = 2 πn + π - 7 - \frac{π}{2}

x = \frac{4 πn + π - 10}{10}, x = 2 πn + π - 7 - \frac{π}{2}

作图

流行的例子

(sin(b))/(300)=(sin(49))/(500)

\frac{sin ( b )}{300} = \frac{sin ( 4 9 ^{\circ} )}{500}

sin(θ)=cos(2θ+60)

sin (θ) = cos (2 θ + 6 0^{\circ})

tan(θ)=(-12)/5

tan (θ) = \frac{- 12}{5}

(1+tan(x/2))/(1-tan(x/2))=sqrt(4.137131)

\frac{1 + tan ( \frac{x}{2} )}{1 - tan ( \frac{x}{2} )} = 4.137131

cos (θ) = 0.51