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受欢迎的三角函数 >

tan(3x)+1=sec(3x)

解答

tan (3 x) + 1 = sec (3 x)

解答

x = \frac{2 πn}{3}

+1

度数

x = 0^{\circ} + 12 0^{\circ} n

求解步骤

tan (3 x) + 1 = sec (3 x)

两边减去

sec (3 x)

tan (3 x) + 1 - sec (3 x) = 0

用 sin, cos 表示

1 - sec (3 x) + tan (3 x)

使用基本三角恒等式:

sec (x) = \frac{1}{cos ( x )}

= 1 - \frac{1}{cos ( 3 x )} + tan (3 x)

使用基本三角恒等式:

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

= 1 - \frac{1}{cos ( 3 x )} + \frac{sin ( 3 x )}{cos ( 3 x )}

化简

1 - \frac{1}{cos ( 3 x )} + \frac{sin ( 3 x )}{cos ( 3 x )} : \frac{cos ( 3 x ) - 1 + sin ( 3 x )}{cos ( 3 x )}

1 - \frac{1}{cos ( 3 x )} + \frac{sin ( 3 x )}{cos ( 3 x )}

合并分式

- \frac{1}{cos ( 3 x )} + \frac{sin ( 3 x )}{cos ( 3 x )} : \frac{- 1 + sin ( 3 x )}{cos ( 3 x )}

使用法则

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

= \frac{- 1 + sin ( 3 x )}{cos ( 3 x )}

= 1 + \frac{sin ( 3 x ) - 1}{cos ( 3 x )}

将项转换为分式:

1 = \frac{1 cos ( 3 x )}{cos ( 3 x )}

= \frac{1 \cdot cos ( 3 x )}{cos ( 3 x )} + \frac{- 1 + sin ( 3 x )}{cos ( 3 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{1 \cdot cos ( 3 x ) - 1 + sin ( 3 x )}{cos ( 3 x )}

乘以:

1 \cdot cos (3 x) = cos (3 x)

= \frac{cos ( 3 x ) - 1 + sin ( 3 x )}{cos ( 3 x )}

= \frac{cos ( 3 x ) - 1 + sin ( 3 x )}{cos ( 3 x )}

\frac{- 1 + cos ( 3 x ) + sin ( 3 x )}{cos ( 3 x )} = 0

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

- 1 + cos (3 x) + sin (3 x) = 0

使用三角恒等式改写

- 1 + cos (3 x) + sin (3 x)

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

sin (3 x) + cos (3 x)

改写为

= 2 (\frac{1}{2} sin (3 x) + \frac{1}{2} cos (3 x))

使用以下普通恒等式:

cos (\frac{π}{4}) = \frac{1}{2}

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{1}{2}

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

使用角和恒等式:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (3 x + \frac{π}{4})

= - 1 + 2 sin (3 x + \frac{π}{4})

- 1 + 2 sin (3 x + \frac{π}{4}) = 0

将

1

到右边

- 1 + 2 sin (3 x + \frac{π}{4}) = 0

两边加上

1

- 1 + 2 sin (3 x + \frac{π}{4}) + 1 = 0 + 1

化简

2 sin (3 x + \frac{π}{4}) = 1

2 sin (3 x + \frac{π}{4}) = 1

两边除以

2

2 sin (3 x + \frac{π}{4}) = 1

两边除以

2

\frac{2 sin ( 3 x + \frac{π}{4} )}{2} = \frac{1}{2}

化简

\frac{2 sin ( 3 x + \frac{π}{4} )}{2} = \frac{1}{2}

化简

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

\frac{2 sin ( 3 x + \frac{π}{4} )}{2}

约分:

2

= sin (3 x + \frac{π}{4})

化简

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

乘以共轭根式

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (3 x + \frac{π}{4}) = \frac{2}{2}

sin (3 x + \frac{π}{4}) = \frac{2}{2}

sin (3 x + \frac{π}{4}) = \frac{2}{2}

sin (3 x + \frac{π}{4}) = \frac{2}{2}

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}

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

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

解

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

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

两边减去

\frac{π}{4}

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

化简

3 x = 2 πn

两边除以

3

3 x = 2 πn

两边除以

3

\frac{3 x}{3} = \frac{2 πn}{3}

化简

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

解

3 x + \frac{π}{4} = \frac{3 π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{π}{6}

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

将

\frac{π}{4}

到右边

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

两边减去

\frac{π}{4}

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

化简

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

化简

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

化简

\frac{3 π}{4} + 2 πn - \frac{π}{4} : 2 πn + \frac{π}{2}

\frac{3 π}{4} + 2 πn - \frac{π}{4}

对同类项分组

= 2 πn - \frac{π}{4} + \frac{3 π}{4}

合并分式

- \frac{π}{4} + \frac{3 π}{4} : \frac{π}{2}

使用法则

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

= \frac{- π + 3 π}{4}

同类项相加:

- π + 3 π = 2 π

= \frac{2 π}{4}

约分:

2

= \frac{π}{2}

= 2 πn + \frac{π}{2}

3 x = 2 πn + \frac{π}{2}

3 x = 2 πn + \frac{π}{2}

3 x = 2 πn + \frac{π}{2}

两边除以

3

3 x = 2 πn + \frac{π}{2}

两边除以

3

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

化简

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

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{2 πn}{3} + \frac{\frac{π}{2}}{3} : \frac{2 πn}{3} + \frac{π}{6}

\frac{2 πn}{3} + \frac{\frac{π}{2}}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

使用分式法则:

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

= \frac{π}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{2 πn}{3} + \frac{π}{6}

x = \frac{2 πn}{3} + \frac{π}{6}

x = \frac{2 πn}{3} + \frac{π}{6}

x = \frac{2 πn}{3} + \frac{π}{6}

x = \frac{2 πn}{3}, x = \frac{2 πn}{3} + \frac{π}{6}

因为方程对以下值无定义:

\frac{2 πn}{3} + \frac{π}{6}

x = \frac{2 πn}{3}

作图

流行的例子

-2sin^2(θ)+sin(θ)+1=0

- 2 sin^{2} (θ) + sin (θ) + 1 = 0

cot (θ) = \frac{1}{4}

sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ)

3 sin (θ) cot (θ) - 2 sin (θ) = sin (θ)

4-7sin^2(θ)=3cos^2(θ)

4 - 7 sin^{2} (θ) = 3 cos^{2} (θ)

csc(3x)=-(2sqrt(3))/3

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