zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的 >

7tan(4x+199)=8sin(4x+199)

解答

7 tan (4 x + 199) = 8 sin (4 x + 199)

解答

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{0.50536 \dots}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{0.50536 \dots}{4}

+1

度数

x = - 2850.46503 \dots^{\circ} + 9 0^{\circ} n, x = - 2805.46503 \dots^{\circ} + 9 0^{\circ} n, x = - 2843.22627 \dots^{\circ} + 9 0^{\circ} n, x = - 2767.70378 \dots^{\circ} + 9 0^{\circ} n

求解步骤

7 tan (4 x + 199) = 8 sin (4 x + 199)

两边减去

8 sin (4 x + 199)

7 tan (4 x + 199) - 8 sin (4 x + 199) = 0

用 sin, cos 表示

7 tan (199 + 4 x) - 8 sin (199 + 4 x)

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= 7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} - 8 sin (199 + 4 x)

化简

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} - 8 sin (199 + 4 x) : \frac{7 sin ( 199 + 4 x ) - 8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} - 8 sin (199 + 4 x)

乘

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )} : \frac{7 sin ( 4 x + 199 )}{cos ( 199 + 4 x )}

7 \cdot \frac{sin ( 199 + 4 x )}{cos ( 199 + 4 x )}

分式相乘:

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

= \frac{sin ( 199 + 4 x ) \cdot 7}{cos ( 199 + 4 x )}

= \frac{7 sin ( 4 x + 199 )}{cos ( 4 x + 199 )} - 8 sin (4 x + 199)

将项转换为分式:

8 sin (4 x + 199) = \frac{8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

= \frac{sin ( 199 + 4 x ) \cdot 7}{cos ( 199 + 4 x )} - \frac{8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

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{sin ( 199 + 4 x ) \cdot 7 - 8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

= \frac{7 sin ( 199 + 4 x ) - 8 sin ( 199 + 4 x ) cos ( 199 + 4 x )}{cos ( 199 + 4 x )}

\frac{7 sin ( 199 + 4 x ) - 8 cos ( 199 + 4 x ) sin ( 199 + 4 x )}{cos ( 199 + 4 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x) = 0

分解

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x) : - sin (199 + 4 x) (8 cos (199 + 4 x) - 7)

7 sin (199 + 4 x) - 8 cos (199 + 4 x) sin (199 + 4 x)

因式分解出通项

- sin (199 + 4 x)

= - sin (199 + 4 x) (- 7 + 8 cos (199 + 4 x))

- sin (199 + 4 x) (8 cos (199 + 4 x) - 7) = 0

分别求解每个部分

sin (199 + 4 x) = 0 or 8 cos (199 + 4 x) - 7 = 0

sin (199 + 4 x) = 0 : x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

sin (199 + 4 x) = 0

sin (199 + 4 x) = 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}

199 + 4 x = 0 + 2 πn, 199 + 4 x = π + 2 πn

199 + 4 x = 0 + 2 πn, 199 + 4 x = π + 2 πn

解

199 + 4 x = 0 + 2 πn : x = \frac{πn}{2} - \frac{199}{4}

199 + 4 x = 0 + 2 πn

0 + 2 πn = 2 πn

199 + 4 x = 2 πn

将

199

到右边

199 + 4 x = 2 πn

两边减去

199

199 + 4 x - 199 = 2 πn - 199

化简

4 x = 2 πn - 199

4 x = 2 πn - 199

两边除以

4

4 x = 2 πn - 199

两边除以

4

\frac{4 x}{4} = \frac{2 πn}{4} - \frac{199}{4}

化简

x = \frac{πn}{2} - \frac{199}{4}

x = \frac{πn}{2} - \frac{199}{4}

解

199 + 4 x = π + 2 πn : x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

199 + 4 x = π + 2 πn

将

199

到右边

199 + 4 x = π + 2 πn

两边减去

199

199 + 4 x - 199 = π + 2 πn - 199

化简

4 x = π + 2 πn - 199

4 x = π + 2 πn - 199

两边除以

4

4 x = π + 2 πn - 199

两边除以

4

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4}

化简

\frac{4 x}{4} = \frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4}

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4} : \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

\frac{π}{4} + \frac{2 πn}{4} - \frac{199}{4}

对同类项分组

= \frac{π}{4} - \frac{199}{4} + \frac{2 πn}{4}

消掉

\frac{2 πn}{4} : \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

= \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}

8 cos (199 + 4 x) - 7 = 0 : x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

8 cos (199 + 4 x) - 7 = 0

将

7

到右边

8 cos (199 + 4 x) - 7 = 0

两边加上

7

8 cos (199 + 4 x) - 7 + 7 = 0 + 7

化简

8 cos (199 + 4 x) = 7

8 cos (199 + 4 x) = 7

两边除以

8

8 cos (199 + 4 x) = 7

两边除以

8

\frac{8 cos ( 199 + 4 x )}{8} = \frac{7}{8}

化简

cos (199 + 4 x) = \frac{7}{8}

cos (199 + 4 x) = \frac{7}{8}

使用反三角函数性质

cos (199 + 4 x) = \frac{7}{8}

cos (199 + 4 x) = \frac{7}{8}

的通解

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

199 + 4 x = arccos (\frac{7}{8}) + 2 πn, 199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

199 + 4 x = arccos (\frac{7}{8}) + 2 πn, 199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

解

199 + 4 x = arccos (\frac{7}{8}) + 2 πn : x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

199 + 4 x = arccos (\frac{7}{8}) + 2 πn

将

199

到右边

199 + 4 x = arccos (\frac{7}{8}) + 2 πn

两边减去

199

199 + 4 x - 199 = arccos (\frac{7}{8}) + 2 πn - 199

化简

4 x = arccos (\frac{7}{8}) + 2 πn - 199

4 x = arccos (\frac{7}{8}) + 2 πn - 199

两边除以

4

4 x = arccos (\frac{7}{8}) + 2 πn - 199

两边除以

4

\frac{4 x}{4} = \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

化简

\frac{4 x}{4} = \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4} : - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

\frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

对同类项分组

= - \frac{199}{4} + \frac{2 πn}{4} + \frac{arccos ( \frac{7}{8} )}{4}

消掉

\frac{2 πn}{4} : \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

= - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}

解

199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn : x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

将

199

到右边

199 + 4 x = 2 π - arccos (\frac{7}{8}) + 2 πn

两边减去

199

199 + 4 x - 199 = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

化简

4 x = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

4 x = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

两边除以

4

4 x = 2 π - arccos (\frac{7}{8}) + 2 πn - 199

两边除以

4

\frac{4 x}{4} = \frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

化简

\frac{4 x}{4} = \frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4} : \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

\frac{2 π}{4} - \frac{arccos ( \frac{7}{8} )}{4} + \frac{2 πn}{4} - \frac{199}{4}

对同类项分组

= \frac{2 π}{4} - \frac{199}{4} + \frac{2 πn}{4} - \frac{arccos ( \frac{7}{8} )}{4}

消掉

\frac{2 π}{4} : \frac{π}{2}

\frac{2 π}{4}

约分:

2

= \frac{π}{2}

= \frac{π}{2} - \frac{199}{4} + \frac{2 πn}{4} - \frac{arccos ( \frac{7}{8} )}{4}

消掉

\frac{2 πn}{4} : \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

= \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

合并所有解

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{arccos ( \frac{7}{8} )}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{arccos ( \frac{7}{8} )}{4}

以小数形式表示解

x = \frac{πn}{2} - \frac{199}{4}, x = \frac{π}{4} - \frac{199}{4} + \frac{πn}{2}, x = - \frac{199}{4} + \frac{πn}{2} + \frac{0.50536 \dots}{4}, x = \frac{π}{2} - \frac{199}{4} + \frac{πn}{2} - \frac{0.50536 \dots}{4}

流行的例子

4sin^2(x)+4cos(x)-5=0,0<= x<= 2pi

4 sin^{2} (x) + 4 cos (x) - 5 = 0, 0 \leq x \leq 2 π

sqrt(2)sin(2θ)=1

2 sin (2 θ) = 1

cos(θ)= 1/(sqrt(2)),0<= θ<= 4pi

cos (θ) = \frac{1}{2}, 0 \leq θ \leq 4 π

sin (x) a = \frac{π}{2}

tan^{2} (3 θ) = 1