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受欢迎的三角函数 >

cos(x)-cos(2x)=0,0<= x<360

解答

cos (x) - cos (2 x) = 0, 0^{\circ} \leq x < 36 0^{\circ}

解答

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

+1

弧度

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

求解步骤

cos (x) - cos (2 x) = 0, 0^{\circ} \leq x < 36 0^{\circ}

使用三角恒等式改写

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

化简

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

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

同类项相加:

x + 2 x = 3 x

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

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

\frac{x - 2 x}{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}

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

使用负角恒等式:

sin (- x) = - sin (x)

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

使用法则

- (- a) = a

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

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

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

分别求解每个部分

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

sin (\frac{3 x}{2}) = 0, 0 \leq x < 36 0^{\circ} : x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

sin (\frac{3 x}{2}) = 0, 0 \leq x < 36 0^{\circ}

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

的通解

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} 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}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{3 x}{2} = 0 + 36 0^{\circ} n, \frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

解

\frac{3 x}{2} = 0 + 36 0^{\circ} n : x = \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

\frac{3 x}{2} = 36 0^{\circ} n

在两边乘以

2

\frac{3 x}{2} = 36 0^{\circ} n

在两边乘以

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 36 0^{\circ} n

化简

3 x = 72 0^{\circ} n

3 x = 72 0^{\circ} n

两边除以

3

3 x = 72 0^{\circ} n

两边除以

3

\frac{3 x}{3} = \frac{72 0 ^{\circ} n}{3}

化简

x = \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}

解

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n : x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

在两边乘以

2

\frac{3 x}{2} = 18 0^{\circ} + 36 0^{\circ} n

在两边乘以

2

\frac{2 \cdot 3 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

化简

3 x = 36 0^{\circ} + 72 0^{\circ} n

3 x = 36 0^{\circ} + 72 0^{\circ} n

两边除以

3

3 x = 36 0^{\circ} + 72 0^{\circ} n

两边除以

3

\frac{3 x}{3} = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

化简

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

x = \frac{72 0 ^{\circ} n}{3}, x = 12 0^{\circ} + \frac{72 0 ^{\circ} n}{3}

在

0 \leq x < 36 0^{\circ}

范围内的解

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

sin (\frac{x}{2}) = 0, 0 \leq x < 36 0^{\circ} : x = 0

sin (\frac{x}{2}) = 0, 0 \leq x < 36 0^{\circ}

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

的通解

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n, \frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

解

\frac{x}{2} = 0 + 36 0^{\circ} n : x = 72 0^{\circ} n

\frac{x}{2} = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

\frac{x}{2} = 36 0^{\circ} n

在两边乘以

2

\frac{x}{2} = 36 0^{\circ} n

在两边乘以

2

\frac{2 x}{2} = 2 \cdot 36 0^{\circ} n

化简

x = 72 0^{\circ} n

x = 72 0^{\circ} n

解

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} + 72 0^{\circ} n

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

在两边乘以

2

\frac{x}{2} = 18 0^{\circ} + 36 0^{\circ} n

在两边乘以

2

\frac{2 x}{2} = 36 0^{\circ} + 2 \cdot 36 0^{\circ} n

化简

x = 36 0^{\circ} + 72 0^{\circ} n

x = 36 0^{\circ} + 72 0^{\circ} n

x = 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

在

0 \leq x < 36 0^{\circ}

范围内的解

x = 0

合并所有解

x = 0, x = 12 0^{\circ}, x = 24 0^{\circ}

作图

流行的例子

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2cos(θ)-sqrt(3)=0,0<= θ<= 2pi

2 cos (θ) - 3 = 0, 0 \leq θ \leq 2 π

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tan (2 x) cos (2 x) + cot (2 x) sin (2 x) = 1

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