English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

1sin(x)=1.5sin(x/2)

解答

1 \cdot sin (x) = 1.5 \cdot sin (\frac{x}{2})

解答

x = 4 πn, x = 2 π (2 n + 1), x = \frac{0.72273 \dots + 2 πn}{0.5}, x = \frac{2 π - 0.72273 \dots + 2 πn}{0.5}

度数

x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n, x = 82.81924 \dots^{\circ} + 72 0^{\circ} n, x = 637.18075 \dots^{\circ} + 72 0^{\circ} n

求解步骤

1 \cdot sin (x) = 1.5 sin (\frac{x}{2})

两边减去

1.5 sin (\frac{x}{2})

sin (x) - 1.5 sin (0.5 x) = 0

令

: u = 0.5 x

sin (2 u) - 1.5 sin (u) = 0

使用三角恒等式改写

sin (2 u) - 1.5 sin (u)

使用倍角公式:

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

= 2 sin (u) cos (u) - 1.5 sin (u)

- 1.5 sin (u) + 2 cos (u) sin (u) = 0

分解

- 1.5 sin (u) + 2 cos (u) sin (u) : sin (u) (2 cos (u) - 1.5)

- 1.5 sin (u) + 2 cos (u) sin (u)

因式分解出通项

sin (u)

= sin (u) (- 1.5 + 2 cos (u))

sin (u) (2 cos (u) - 1.5) = 0

分别求解每个部分

sin (u) = 0 or 2 cos (u) - 1.5 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

解

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

2 cos (u) - 1.5 = 0 : u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

2 cos (u) - 1.5 = 0

将

1.5

到右边

2 cos (u) - 1.5 = 0

两边加上

1.5

2 cos (u) - 1.5 + 1.5 = 0 + 1.5

化简

2 cos (u) = 1.5

2 cos (u) = 1.5

两边除以

2

2 cos (u) = 1.5

两边除以

2

\frac{2 cos ( u )}{2} = \frac{1.5}{2}

化简

cos (u) = 0.75

cos (u) = 0.75

使用反三角函数性质

cos (u) = 0.75

cos (u) = 0.75

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

合并所有解

u = 2 πn, u = π + 2 πn, u = arccos (0.75) + 2 πn, u = 2 π - arccos (0.75) + 2 πn

u = 0.5 x

代回

0.5 x = 2 πn : x = 4 πn

0.5 x = 2 πn

在两边乘以

10

0.5 x = 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 x \cdot 10 = 2 πn \cdot 10

整理后得

5 x = 20 πn

5 x = 20 πn

两边除以

5

5 x = 20 πn

两边除以

5

\frac{5 x}{5} = \frac{20 πn}{5}

化简

x = 4 πn

x = 4 πn

0.5 x = π + 2 πn : x = 2 π (2 n + 1)

0.5 x = π + 2 πn

在两边乘以

10

0.5 x = π + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 x \cdot 10 = π 10 + 2 πn \cdot 10

整理后得

5 x = 10 π + 20 πn

5 x = 10 π + 20 πn

两边除以

5

5 x = 10 π + 20 πn

两边除以

5

\frac{5 x}{5} = \frac{10 π}{5} + \frac{20 πn}{5}

化简

\frac{5 x}{5} = \frac{10 π}{5} + \frac{20 πn}{5}

化简

\frac{5 x}{5} : x

\frac{5 x}{5}

数字相除:

\frac{5}{5} = 1

= x

化简

\frac{10 π}{5} + \frac{20 πn}{5} : 2 π (2 n + 1)

\frac{10 π}{5} + \frac{20 πn}{5}

使用法则

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

= \frac{10 π + 20 πn}{5}

分解

10 π + 20 πn : 10 π (1 + 2 n)

10 π + 20 πn

改写为

= 1 \cdot 10 π + 2 \cdot 10 πn

因式分解出通项

10 π

= 10 π (1 + 2 n)

= \frac{10 π ( 1 + 2 n )}{5}

数字相除:

\frac{10}{5} = 2

= 2 π (2 n + 1)

x = 2 π (2 n + 1)

x = 2 π (2 n + 1)

x = 2 π (2 n + 1)

0.5 x = arccos (0.75) + 2 πn : x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

0.5 x = arccos (0.75) + 2 πn

两边除以

0.5

0.5 x = arccos (0.75) + 2 πn

两边除以

0.5

\frac{0.5 x}{0.5} = \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

化简

\frac{0.5 x}{0.5} = \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

化简

\frac{0.5 x}{0.5} : x

\frac{0.5 x}{0.5}

约分:

0.5

= x

化简

\frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5} : \frac{arccos ( 0.75 ) + 2 πn}{0.5}

\frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

使用法则

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

= \frac{arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}

0.5 x = 2 π - arccos (0.75) + 2 πn : x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

0.5 x = 2 π - arccos (0.75) + 2 πn

两边除以

0.5

0.5 x = 2 π - arccos (0.75) + 2 πn

两边除以

0.5

\frac{0.5 x}{0.5} = \frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

化简

\frac{0.5 x}{0.5} = \frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

化简

\frac{0.5 x}{0.5} : x

\frac{0.5 x}{0.5}

约分:

0.5

= x

化简

\frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5} : \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

\frac{2 π}{0.5} - \frac{arccos ( 0.75 )}{0.5} + \frac{2 πn}{0.5}

使用法则

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

= \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

x = 4 πn, x = 2 π (2 n + 1), x = \frac{arccos ( 0.75 ) + 2 πn}{0.5}, x = \frac{2 π - arccos ( 0.75 ) + 2 πn}{0.5}

以小数形式表示解

x = 4 πn, x = 2 π (2 n + 1), x = \frac{0.72273 \dots + 2 πn}{0.5}, x = \frac{2 π - 0.72273 \dots + 2 πn}{0.5}

作图

流行的例子

(sin(90))/5 =(sin(x))/4

\frac{sin ( 9 0 ^{\circ} )}{5} = \frac{sin ( x )}{4}

sin(x)=-0.2672

sin (x) = - 0.2672

sin(x)= 3/(sqrt(10))

sin (x) = \frac{3}{10}

-20cos(θ)+10=0

- 20 cos (θ) + 10 = 0

sin(x)-1=-1

sin (x) - 1 = - 1

1*sin(x)=1.5*sin(x/2)

解答

解答

作图

流行的例子

1sin(x)=1.5sin(x/2)