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受欢迎的三角函数 >

solvefor x,sin(x+r/4)+sin(x-r/3)=0

解答

求解

x, sin (x + \frac{r}{4}) + sin (x - \frac{r}{3}) = 0

解答

x = 2 πn + \frac{r}{24}, x = π + 2 πn + \frac{r}{24}

求解步骤

sin (x + \frac{r}{4}) + sin (x - \frac{r}{3}) = 0

使用三角恒等式改写

sin (x + \frac{r}{4}) + sin (x - \frac{r}{3})

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{x + \frac{r}{4} + x - \frac{r}{3}}{2}) cos (\frac{x + \frac{r}{4} - ( x - \frac{r}{3} )}{2})

化简

2 sin (\frac{x + \frac{r}{4} + x - \frac{r}{3}}{2}) cos (\frac{x + \frac{r}{4} - ( x - \frac{r}{3} )}{2}) : 2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24})

2 sin (\frac{x + \frac{r}{4} + x - \frac{r}{3}}{2}) cos (\frac{x + \frac{r}{4} - ( x - \frac{r}{3} )}{2})

\frac{x + \frac{r}{4} + x - \frac{r}{3}}{2} = \frac{24 x - r}{24}

\frac{x + \frac{r}{4} + x - \frac{r}{3}}{2}

x + \frac{r}{4} + x - \frac{r}{3} = 2 x + \frac{r}{4} - \frac{r}{3}

x + \frac{r}{4} + x - \frac{r}{3}

对同类项分组

= x + x + \frac{r}{4} - \frac{r}{3}

同类项相加:

x + x = 2 x

= 2 x + \frac{r}{4} - \frac{r}{3}

= \frac{2 x + \frac{r}{4} - \frac{r}{3}}{2}

化简

2 x + \frac{r}{4} - \frac{r}{3} : \frac{24 x - r}{12}

2 x + \frac{r}{4} - \frac{r}{3}

将项转换为分式:

2 x = \frac{2 x}{1}

= \frac{2 x}{1} + \frac{r}{4} - \frac{r}{3}

1, 4, 3

的最小公倍数:

12

1, 4, 3

最小公倍数 (LCM)

1

质因数分解

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

计算出由至少在以下一个数字中出现的因数组成的数字:

1, 4, 3

= 2 \cdot 2 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 = 12

= 12

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

12

对于

\frac{2 x}{1} :

将分母和分子乘以

12

\frac{2 x}{1} = \frac{2 x \cdot 12}{1 \cdot 12} = \frac{24 x}{12}

对于

\frac{r}{4} :

将分母和分子乘以

3

\frac{r}{4} = \frac{r \cdot 3}{4 \cdot 3} = \frac{r \cdot 3}{12}

对于

\frac{r}{3} :

将分母和分子乘以

4

\frac{r}{3} = \frac{r \cdot 4}{3 \cdot 4} = \frac{r \cdot 4}{12}

= \frac{24 x}{12} + \frac{r \cdot 3}{12} - \frac{r \cdot 4}{12}

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{24 x + r \cdot 3 - r \cdot 4}{12}

同类项相加:

3 r - 4 r = - r

= \frac{24 x - r}{12}

= \frac{\frac{24 x - r}{12}}{2}

使用分式法则:

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

= \frac{24 x - r}{12 \cdot 2}

数字相乘:

12 \cdot 2 = 24

= \frac{24 x - r}{24}

= 2 sin (\frac{24 x - r}{24}) cos (\frac{x - ( x - \frac{r}{3} ) + \frac{r}{4}}{2})

\frac{x + \frac{r}{4} - ( x - \frac{r}{3} )}{2} = \frac{7 r}{24}

\frac{x + \frac{r}{4} - ( x - \frac{r}{3} )}{2}

化简

x + \frac{r}{4} - (x - \frac{r}{3}) : \frac{7 r}{12}

x + \frac{r}{4} - (x - \frac{r}{3})

将项转换为分式:

x = \frac{x 4}{4}, (x - \frac{r}{3}) = \frac{( x - \frac{r}{3} ) 4}{4}

= \frac{x \cdot 4}{4} + \frac{r}{4} - \frac{( x - \frac{r}{3} ) \cdot 4}{4}

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{x \cdot 4 + r - ( x - \frac{r}{3} ) \cdot 4}{4}

乘开

x \cdot 4 + r - (x - \frac{r}{3}) \cdot 4 : \frac{4 r}{3} + r

x \cdot 4 + r - (x - \frac{r}{3}) \cdot 4

= 4 x + r - 4 (x - \frac{r}{3})

乘开

- 4 (x - \frac{r}{3}) : - 4 x + \frac{4 r}{3}

- 4 (x - \frac{r}{3})

使用分配律:

a (b - c) = ab - a c

a = - 4, b = x, c = \frac{r}{3}

= - 4 x - (- 4) \frac{r}{3}

使用加减运算法则

- (- a) = a

= - 4 x + 4 \cdot \frac{r}{3}

乘

4 \cdot \frac{r}{3} : \frac{4 r}{3}

4 \cdot \frac{r}{3}

分式相乘:

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

= \frac{r \cdot 4}{3}

= - 4 x + \frac{4 r}{3}

= x \cdot 4 + r - 4 x + \frac{4 r}{3}

化简

x \cdot 4 + r - 4 x + \frac{4 r}{3} : \frac{4 r}{3} + r

x \cdot 4 + r - 4 x + \frac{4 r}{3}

对同类项分组

= 4 x - 4 x + \frac{4 r}{3} + r

同类项相加:

4 x - 4 x = 0

= \frac{4 r}{3} + r

= \frac{4 r}{3} + r

= \frac{\frac{4 r}{3} + r}{4}

化简

\frac{4 r}{3} + r : \frac{7 r}{3}

\frac{4 r}{3} + r

将项转换为分式:

r = \frac{r 3}{3}

= \frac{4 r}{3} + \frac{r \cdot 3}{3}

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{4 r + r \cdot 3}{3}

同类项相加:

4 r + 3 r = 7 r

= \frac{7 r}{3}

= \frac{\frac{7 r}{3}}{4}

使用分式法则:

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

= \frac{7 r}{3 \cdot 4}

数字相乘:

3 \cdot 4 = 12

= \frac{7 r}{12}

= \frac{\frac{7 r}{12}}{2}

使用分式法则:

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

= \frac{7 r}{12 \cdot 2}

数字相乘:

12 \cdot 2 = 24

= \frac{7 r}{24}

= 2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24})

= 2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24})

2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24}) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

sin (\frac{24 x - r}{24}) = 0

sin (\frac{24 x - r}{24}) = 0

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{24 x - r}{24} = 0 + 2 πn, \frac{24 x - r}{24} = π + 2 πn

\frac{24 x - r}{24} = 0 + 2 πn, \frac{24 x - r}{24} = π + 2 πn

解

\frac{24 x - r}{24} = 0 + 2 πn : x = 2 πn + \frac{r}{24}

\frac{24 x - r}{24} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{24 x - r}{24} = 2 πn

在两边乘以

24

\frac{24 x - r}{24} = 2 πn

在两边乘以

24

\frac{24 ( 24 x - r )}{24} = 24 \cdot 2 πn

化简

24 x - r = 48 πn

24 x - r = 48 πn

将

r

到右边

24 x - r = 48 πn

两边加上

r

24 x - r + r = 48 πn + r

化简

24 x = 48 πn + r

24 x = 48 πn + r

两边除以

24

24 x = 48 πn + r

两边除以

24

\frac{24 x}{24} = \frac{48 πn}{24} + \frac{r}{24}

化简

x = 2 πn + \frac{r}{24}

x = 2 πn + \frac{r}{24}

解

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

\frac{24 x - r}{24} = π + 2 πn

在两边乘以

24

\frac{24 x - r}{24} = π + 2 πn

在两边乘以

24

\frac{24 ( 24 x - r )}{24} = 24 π + 24 \cdot 2 πn

化简

24 x - r = 24 π + 48 πn

24 x - r = 24 π + 48 πn

将

r

到右边

24 x - r = 24 π + 48 πn

两边加上

r

24 x - r + r = 24 π + 48 πn + r

化简

24 x = 24 π + 48 πn + r

24 x = 24 π + 48 πn + r

两边除以

24

24 x = 24 π + 48 πn + r

两边除以

24

\frac{24 x}{24} = \frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24}

化简

\frac{24 x}{24} = \frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24}

化简

\frac{24 x}{24} : x

\frac{24 x}{24}

数字相除:

\frac{24}{24} = 1

= x

化简

\frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24} : π + 2 πn + \frac{r}{24}

\frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24}

数字相除:

\frac{24}{24} = 1

= π + \frac{48 πn}{24} + \frac{r}{24}

数字相除:

\frac{48}{24} = 2

= π + 2 πn + \frac{r}{24}

x = π + 2 πn + \frac{r}{24}

x = π + 2 πn + \frac{r}{24}

x = π + 2 πn + \frac{r}{24}

x = 2 πn + \frac{r}{24}, x = π + 2 πn + \frac{r}{24}

作图

流行的例子

cos^2(x)=2sin^2(x)

cos^{2} (x) = 2 sin^{2} (x)

solvefor x,sec(x)-sin(50o)=2

so l v e f or x, sec (x) - sin (50 o) = 2

2sin^2(x)-2cos^2(x)=0

2 sin^{2} (x) - 2 cos^{2} (x) = 0

sin^2(x)+2cos^2(x)=2

sin^{2} (x) + 2 cos^{2} (x) = 2

solvefor x,cos^2(x)+cos^2(y)cos^2(z)=1

so l v e f or x, cos^{2} (x) + cos^{2} (y) cos^{2} (z) = 1