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受欢迎的三角函数 >

sin(10x)=sin(10(x+1))

解答

sin (10 x) = sin (10 (x + 1))

解答

x = - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}, x = - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

+1

度数

x = - 19.64788 \dots^{\circ} + 3 6^{\circ} n, x = - 1.64788 \dots^{\circ} + 3 6^{\circ} n

求解步骤

sin (10 x) = sin (10 (x + 1))

两边减去

sin (10 (x + 1))

sin (10 x) - sin (10 (x + 1)) = 0

使用三角恒等式改写

sin (10 x) - sin (10 (x + 1))

使用和差化积恒等式:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

= 2 sin (\frac{10 x - 10 ( x + 1 )}{2}) cos (\frac{10 x + 10 ( x + 1 )}{2})

化简

2 sin (\frac{10 x - 10 ( x + 1 )}{2}) cos (\frac{10 x + 10 ( x + 1 )}{2}) : - 2 sin (5) cos (5 (2 x + 1))

2 sin (\frac{10 x - 10 ( x + 1 )}{2}) cos (\frac{10 x + 10 ( x + 1 )}{2})

\frac{10 x - 10 ( x + 1 )}{2} = - 5

\frac{10 x - 10 ( x + 1 )}{2}

分解

10 x - 10 (x + 1) : - 10

10 x - 10 (x + 1)

因式分解出通项

10

= 10 (x - (1 + x))

乘开

x - (x + 1) : - 1

x - (1 + x)

- (1 + x) : - 1 - x

- (1 + x)

打开括号

= - (1) - (x)

使用加减运算法则

+ (- a) = - a

= - 1 - x

= x - 1 - x

化简

x - 1 - x : - 1

x - 1 - x

对同类项分组

= x - x - 1

同类项相加:

x - x = 0

= - 1

= - 1

= 10 (- 1)

整理后得

= - 10

= - \frac{10}{2}

数字相除:

\frac{10}{2} = 5

= - 5

= 2 sin (- 5) cos (\frac{10 x + 10 ( x + 1 )}{2})

化简

sin (- 5) : - sin (5)

sin (- 5)

利用以下特性:

sin (- x) = - sin (x)

sin (- 5) = - sin (5)

= - sin (5)

= 2 (- sin (5)) cos (\frac{10 x + 10 ( x + 1 )}{2})

\frac{10 x + 10 ( x + 1 )}{2} = 5 (2 x + 1)

\frac{10 x + 10 ( x + 1 )}{2}

分解

10 x + 10 (x + 1) : 10 (2 x + 1)

10 x + 10 (x + 1)

因式分解出通项

10

= 10 (x + 1 + x)

整理后得

= 10 (2 x + 1)

= \frac{10 ( 2 x + 1 )}{2}

数字相除:

\frac{10}{2} = 5

= 5 (2 x + 1)

= 2 (- sin (5)) cos (5 (2 x + 1))

去除括号:

(- a) = - a

= - 2 sin (5) cos (5 (2 x + 1))

= - 2 sin (5) cos (5 (2 x + 1))

- 2 sin (5) cos (5 (2 x + 1)) = 0

两边除以

- 2 sin (5)

- 2 sin (5) cos (5 (2 x + 1)) = 0

两边除以

- 2 sin (5)

\frac{- 2 sin ( 5 ) cos ( 5 ( 2 x + 1 ) )}{- 2 sin ( 5 )} = \frac{0}{- 2 sin ( 5 )}

化简

cos (5 (2 x + 1)) = 0

cos (5 (2 x + 1)) = 0

cos (5 (2 x + 1)) = 0

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

5 (2 x + 1) = \frac{π}{2} + 2 πn, 5 (2 x + 1) = \frac{3 π}{2} + 2 πn

5 (2 x + 1) = \frac{π}{2} + 2 πn, 5 (2 x + 1) = \frac{3 π}{2} + 2 πn

解

5 (2 x + 1) = \frac{π}{2} + 2 πn : x = - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}

5 (2 x + 1) = \frac{π}{2} + 2 πn

两边除以

5

5 (2 x + 1) = \frac{π}{2} + 2 πn

两边除以

5

\frac{5 ( 2 x + 1 )}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5}

化简

\frac{5 ( 2 x + 1 )}{5} = \frac{\frac{π}{2}}{5} + \frac{2 πn}{5}

化简

\frac{5 ( 2 x + 1 )}{5} : 2 x + 1

\frac{5 ( 2 x + 1 )}{5}

数字相除:

\frac{5}{5} = 1

= 2 x + 1

化简

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

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

\frac{\frac{π}{2}}{5} = \frac{π}{10}

\frac{\frac{π}{2}}{5}

使用分式法则:

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

= \frac{π}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{π}{10}

= \frac{π}{10} + \frac{2 πn}{5}

2 x + 1 = \frac{π}{10} + \frac{2 πn}{5}

2 x + 1 = \frac{π}{10} + \frac{2 πn}{5}

2 x + 1 = \frac{π}{10} + \frac{2 πn}{5}

将

1

到右边

2 x + 1 = \frac{π}{10} + \frac{2 πn}{5}

两边减去

1

2 x + 1 - 1 = \frac{π}{10} + \frac{2 πn}{5} - 1

化简

2 x = \frac{π}{10} + \frac{2 πn}{5} - 1

2 x = \frac{π}{10} + \frac{2 πn}{5} - 1

两边除以

2

2 x = \frac{π}{10} + \frac{2 πn}{5} - 1

两边除以

2

\frac{2 x}{2} = \frac{\frac{π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2}

化简

\frac{2 x}{2} = \frac{\frac{π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2} : - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}

\frac{\frac{π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2}

对同类项分组

= - \frac{1}{2} + \frac{\frac{π}{10}}{2} + \frac{\frac{2 πn}{5}}{2}

\frac{\frac{π}{10}}{2} = \frac{π}{20}

\frac{\frac{π}{10}}{2}

使用分式法则:

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

= \frac{π}{10 \cdot 2}

数字相乘:

10 \cdot 2 = 20

= \frac{π}{20}

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

\frac{\frac{2 πn}{5}}{2}

使用分式法则:

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

= \frac{2 πn}{5 \cdot 2}

数字相乘:

5 \cdot 2 = 10

= \frac{2 πn}{10}

约分:

2

= \frac{πn}{5}

= - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}

x = - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}

x = - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}

x = - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}

解

5 (2 x + 1) = \frac{3 π}{2} + 2 πn : x = - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

5 (2 x + 1) = \frac{3 π}{2} + 2 πn

两边除以

5

5 (2 x + 1) = \frac{3 π}{2} + 2 πn

两边除以

5

\frac{5 ( 2 x + 1 )}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5}

化简

\frac{5 ( 2 x + 1 )}{5} = \frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5}

化简

\frac{5 ( 2 x + 1 )}{5} : 2 x + 1

\frac{5 ( 2 x + 1 )}{5}

数字相除:

\frac{5}{5} = 1

= 2 x + 1

化简

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5} : \frac{3 π}{10} + \frac{2 πn}{5}

\frac{\frac{3 π}{2}}{5} + \frac{2 πn}{5}

\frac{\frac{3 π}{2}}{5} = \frac{3 π}{10}

\frac{\frac{3 π}{2}}{5}

使用分式法则:

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

= \frac{3 π}{2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{3 π}{10}

= \frac{3 π}{10} + \frac{2 πn}{5}

2 x + 1 = \frac{3 π}{10} + \frac{2 πn}{5}

2 x + 1 = \frac{3 π}{10} + \frac{2 πn}{5}

2 x + 1 = \frac{3 π}{10} + \frac{2 πn}{5}

将

1

到右边

2 x + 1 = \frac{3 π}{10} + \frac{2 πn}{5}

两边减去

1

2 x + 1 - 1 = \frac{3 π}{10} + \frac{2 πn}{5} - 1

化简

2 x = \frac{3 π}{10} + \frac{2 πn}{5} - 1

2 x = \frac{3 π}{10} + \frac{2 πn}{5} - 1

两边除以

2

2 x = \frac{3 π}{10} + \frac{2 πn}{5} - 1

两边除以

2

\frac{2 x}{2} = \frac{\frac{3 π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2}

化简

\frac{2 x}{2} = \frac{\frac{3 π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{3 π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2} : - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

\frac{\frac{3 π}{10}}{2} + \frac{\frac{2 πn}{5}}{2} - \frac{1}{2}

对同类项分组

= - \frac{1}{2} + \frac{\frac{3 π}{10}}{2} + \frac{\frac{2 πn}{5}}{2}

\frac{\frac{3 π}{10}}{2} = \frac{3 π}{20}

\frac{\frac{3 π}{10}}{2}

使用分式法则:

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

= \frac{3 π}{10 \cdot 2}

数字相乘:

10 \cdot 2 = 20

= \frac{3 π}{20}

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

\frac{\frac{2 πn}{5}}{2}

使用分式法则:

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

= \frac{2 πn}{5 \cdot 2}

数字相乘:

5 \cdot 2 = 10

= \frac{2 πn}{10}

约分:

2

= \frac{πn}{5}

= - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

x = - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

x = - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

x = - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

x = - \frac{1}{2} + \frac{π}{20} + \frac{πn}{5}, x = - \frac{1}{2} + \frac{3 π}{20} + \frac{πn}{5}

作图

流行的例子

sin^2(θ)+3sin(θ)=0

sin^{2} (θ) + 3 sin (θ) = 0

cos (x) = - 0.7962

1.04cos^2(x)+0.102cos(x)-0.935=0

1.04 cos^{2} (x) + 0.102 cos (x) - 0.935 = 0

(sin(88))/(26)=(sin(x))/(21)

\frac{sin ( 8 8 ^{\circ} )}{26} = \frac{sin ( x )}{21}

tan (x) = \frac{62}{21}