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受欢迎的三角函数 >

|cos(x/4)|= 1/2

解答

cos (\frac{x}{4}) = \frac{1}{2}

解答

x = \frac{8 π}{3} + 8 πn, x = \frac{16 π}{3} + 8 πn, x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

+1

度数

x = 48 0^{\circ} + 144 0^{\circ} n, x = 96 0^{\circ} + 144 0^{\circ} n, x = 24 0^{\circ} + 144 0^{\circ} n, x = 120 0^{\circ} + 144 0^{\circ} n

求解步骤

cos (\frac{x}{4}) = \frac{1}{2}

用替代法求解

cos (\frac{x}{4}) = \frac{1}{2}

令

: cos (\frac{x}{4}) = u

∣ u ∣ = \frac{1}{2}

∣ u ∣ = \frac{1}{2} : u = - \frac{1}{2} or u = \frac{1}{2}

∣ u ∣ = \frac{1}{2}

使用绝对值运算法则: 若

∣ u ∣ = a, a > 0

，则

u = a

or

u = - a

u = - \frac{1}{2} or u = \frac{1}{2}

u = cos (\frac{x}{4})

代回

cos (\frac{x}{4}) = - \frac{1}{2} or cos (\frac{x}{4}) = \frac{1}{2}

cos (\frac{x}{4}) = - \frac{1}{2} or cos (\frac{x}{4}) = \frac{1}{2}

cos (\frac{x}{4}) = - \frac{1}{2} : x = \frac{8 π}{3} + 8 πn, x = \frac{16 π}{3} + 8 πn

cos (\frac{x}{4}) = - \frac{1}{2}

cos (\frac{x}{4}) = - \frac{1}{2}

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}

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

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

解

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

\frac{x}{4} = \frac{2 π}{3} + 2 πn

在两边乘以

4

\frac{x}{4} = \frac{2 π}{3} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

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

4 \cdot \frac{2 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{2 π}{3} = \frac{8 π}{3}

4 \cdot \frac{2 π}{3}

分式相乘:

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

= \frac{2 π 4}{3}

数字相乘:

2 \cdot 4 = 8

= \frac{8 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= \frac{8 π}{3} + 8 πn

x = \frac{8 π}{3} + 8 πn

x = \frac{8 π}{3} + 8 πn

x = \frac{8 π}{3} + 8 πn

解

\frac{x}{4} = \frac{4 π}{3} + 2 πn : x = \frac{16 π}{3} + 8 πn

\frac{x}{4} = \frac{4 π}{3} + 2 πn

在两边乘以

4

\frac{x}{4} = \frac{4 π}{3} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

4 \cdot \frac{4 π}{3} + 4 \cdot 2 πn : \frac{16 π}{3} + 8 πn

4 \cdot \frac{4 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{4 π}{3} = \frac{16 π}{3}

4 \cdot \frac{4 π}{3}

分式相乘:

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

= \frac{4 π 4}{3}

数字相乘:

4 \cdot 4 = 16

= \frac{16 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= \frac{16 π}{3} + 8 πn

x = \frac{16 π}{3} + 8 πn

x = \frac{16 π}{3} + 8 πn

x = \frac{16 π}{3} + 8 πn

x = \frac{8 π}{3} + 8 πn, x = \frac{16 π}{3} + 8 πn

cos (\frac{x}{4}) = \frac{1}{2} : x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

cos (\frac{x}{4}) = \frac{1}{2}

cos (\frac{x}{4}) = \frac{1}{2}

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}

\frac{x}{4} = \frac{π}{3} + 2 πn, \frac{x}{4} = \frac{5 π}{3} + 2 πn

\frac{x}{4} = \frac{π}{3} + 2 πn, \frac{x}{4} = \frac{5 π}{3} + 2 πn

解

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

\frac{x}{4} = \frac{π}{3} + 2 πn

在两边乘以

4

\frac{x}{4} = \frac{π}{3} + 2 πn

在两边乘以

4

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

化简

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

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

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

4 \cdot \frac{π}{3} + 4 \cdot 2 πn

乘

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

分式相乘:

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

= \frac{π 4}{3}

= \frac{4 π}{3} + 4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= \frac{4 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn

解

\frac{x}{4} = \frac{5 π}{3} + 2 πn : x = \frac{20 π}{3} + 8 πn

\frac{x}{4} = \frac{5 π}{3} + 2 πn

在两边乘以

4

\frac{x}{4} = \frac{5 π}{3} + 2 πn

在两边乘以

4

\frac{4 x}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

化简

\frac{4 x}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn : \frac{20 π}{3} + 8 πn

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{3} = \frac{20 π}{3}

4 \cdot \frac{5 π}{3}

分式相乘:

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

= \frac{5 π 4}{3}

数字相乘:

5 \cdot 4 = 20

= \frac{20 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= \frac{20 π}{3} + 8 πn

x = \frac{20 π}{3} + 8 πn

x = \frac{20 π}{3} + 8 πn

x = \frac{20 π}{3} + 8 πn

x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

合并所有解

x = \frac{8 π}{3} + 8 πn, x = \frac{16 π}{3} + 8 πn, x = \frac{4 π}{3} + 8 πn, x = \frac{20 π}{3} + 8 πn

作图

流行的例子

cos(2x)=5cos(x)-3

cos (2 x) = 5 cos (x) - 3

4tan(A)=2tan(A)-sqrt(2)

4 tan (A) = 2 tan (A) - 2

2sin(θ)=tan(θ)

2 sin (θ) = tan (θ)

sec (θ) = - 2.5

cos(4x)=0,0<= x<= 2pi

cos (4 x) = 0, 0 \leq x \leq 2 π