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受欢迎的三角函数 >

cos(x/2)=cos(x),0<= x<= 2pi

解答

cos (\frac{x}{2}) = cos (x), 0 \leq x \leq 2 π

解答

x = 0, x = \frac{4 π}{3}

+1

度数

x = 0^{\circ}, x = 24 0^{\circ}

求解步骤

cos (\frac{x}{2}) = cos (x), 0 \leq x \leq 2 π

两边减去

cos (x)

cos (\frac{x}{2}) - cos (x) = 0

使用三角恒等式改写

cos (\frac{x}{2}) - cos (x)

使用和差化积恒等式:

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

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

化简

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

- 2 sin (\frac{\frac{x}{2} + x}{2}) sin (\frac{\frac{x}{2} - x}{2})

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

\frac{\frac{x}{2} + x}{2}

化简

\frac{x}{2} + x : \frac{3 x}{2}

\frac{x}{2} + x

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x}{2} + \frac{x \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{x + x \cdot 2}{2}

同类项相加:

x + 2 x = 3 x

= \frac{3 x}{2}

= \frac{\frac{3 x}{2}}{2}

使用分式法则:

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

= \frac{3 x}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{3 x}{4}

= - 2 sin (\frac{3 x}{4}) sin (\frac{\frac{x}{2} - x}{2})

\frac{\frac{x}{2} - x}{2} = - \frac{x}{4}

\frac{\frac{x}{2} - x}{2}

化简

\frac{x}{2} - x : - \frac{x}{2}

\frac{x}{2} - x

将项转换为分式:

x = \frac{x 2}{2}

= \frac{x}{2} - \frac{x \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{x - x \cdot 2}{2}

同类项相加:

x - 2 x = - x

= \frac{- x}{2}

使用分式法则:

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

= - \frac{x}{2}

= \frac{- \frac{x}{2}}{2}

使用分式法则:

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

= - \frac{\frac{x}{2}}{2}

使用分式法则:

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

\frac{\frac{x}{2}}{2} = \frac{x}{2 \cdot 2}

= - \frac{x}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= - \frac{x}{4}

= - 2 sin (\frac{3 x}{4}) sin (- \frac{x}{4})

使用负角恒等式:

sin (- x) = - sin (x)

= - 2 (- sin (\frac{x}{4})) sin (\frac{3 x}{4})

使用法则

- (- a) = a

= 2 sin (\frac{x}{4}) sin (\frac{3 x}{4})

= 2 sin (\frac{x}{4}) sin (\frac{3 x}{4})

2 sin (\frac{3 x}{4}) sin (\frac{x}{4}) = 0

分别求解每个部分

sin (\frac{3 x}{4}) = 0 or sin (\frac{x}{4}) = 0

sin (\frac{3 x}{4}) = 0, 0 \leq x \leq 2 π : x = 0, x = \frac{4 π}{3}

sin (\frac{3 x}{4}) = 0, 0 \leq x \leq 2 π

sin (\frac{3 x}{4}) = 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{3 x}{4} = 0 + 2 πn, \frac{3 x}{4} = π + 2 πn

\frac{3 x}{4} = 0 + 2 πn, \frac{3 x}{4} = π + 2 πn

解

\frac{3 x}{4} = 0 + 2 πn : x = \frac{8 πn}{3}

\frac{3 x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x}{4} = 2 πn

在两边乘以

4

\frac{3 x}{4} = 2 πn

在两边乘以

4

\frac{4 \cdot 3 x}{4} = 4 \cdot 2 πn

化简

3 x = 8 πn

3 x = 8 πn

两边除以

3

3 x = 8 πn

两边除以

3

\frac{3 x}{3} = \frac{8 πn}{3}

化简

x = \frac{8 πn}{3}

x = \frac{8 πn}{3}

解

\frac{3 x}{4} = π + 2 πn : x = \frac{4 π}{3} + \frac{8 πn}{3}

\frac{3 x}{4} = π + 2 πn

在两边乘以

4

\frac{3 x}{4} = π + 2 πn

在两边乘以

4

\frac{4 \cdot 3 x}{4} = 4 π + 4 \cdot 2 πn

化简

3 x = 4 π + 8 πn

3 x = 4 π + 8 πn

两边除以

3

3 x = 4 π + 8 πn

两边除以

3

\frac{3 x}{3} = \frac{4 π}{3} + \frac{8 πn}{3}

化简

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

在

0 \leq x \leq 2 π

范围内的解

x = 0, x = \frac{4 π}{3}

sin (\frac{x}{4}) = 0, 0 \leq x \leq 2 π : x = 0

sin (\frac{x}{4}) = 0, 0 \leq x \leq 2 π

sin (\frac{x}{4}) = 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{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

\frac{x}{4} = 0 + 2 πn, \frac{x}{4} = π + 2 πn

解

\frac{x}{4} = 0 + 2 πn : x = 8 πn

\frac{x}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{4} = 2 πn

在两边乘以

4

\frac{x}{4} = 2 πn

在两边乘以

4

\frac{4 x}{4} = 4 \cdot 2 πn

化简

x = 8 πn

x = 8 πn

解

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

\frac{x}{4} = π + 2 πn

在两边乘以

4

\frac{x}{4} = π + 2 πn

在两边乘以

4

\frac{4 x}{4} = 4 π + 4 \cdot 2 πn

化简

x = 4 π + 8 πn

x = 4 π + 8 πn

x = 8 πn, x = 4 π + 8 πn

在

0 \leq x \leq 2 π

范围内的解

x = 0

合并所有解

x = 0, x = \frac{4 π}{3}

作图

流行的例子

sin (x) = - \frac{3}{7}

7 - 4 cos (θ) = 7

solvefor t,x=tan^2(t)

so l v e f or t, x = tan^{2} (t)

sin (x) = 0.8246

cos^3(x)+cos(x)=0

cos^{3} (x) + cos (x) = 0