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受欢迎的三角函数 >

1/(cos(x))=tan(x)

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

tan (x)

\frac{1}{cos ( x )} - tan (x) = 0

化简

\frac{1}{cos ( x )} - tan (x) : \frac{1 - tan ( x ) cos ( x )}{cos ( x )}

\frac{1}{cos ( x )} - tan (x)

将项转换为分式:

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

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

\frac{1 - tan ( x ) cos ( x )}{cos ( x )} = 0

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

1 - tan (x) cos (x) = 0

使用三角恒等式改写

1 - tan (x) cos (x)

cos (x) tan (x) = sin (x)

cos (x) tan (x)

使用基本三角恒等式:

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

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

化简

cos (x) \frac{sin ( x )}{cos ( x )} : sin (x)

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

分式相乘:

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

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

约分:

cos (x)

= sin (x)

= sin (x)

= 1 - sin (x)

1 - sin (x) = 0

将

1

到右边

1 - sin (x) = 0

两边减去

1

1 - sin (x) - 1 = 0 - 1

化简

- sin (x) = - 1

- sin (x) = - 1

两边除以

- 1

- sin (x) = - 1

两边除以

- 1

\frac{- sin ( x )}{- 1} = \frac{- 1}{- 1}

化简

sin (x) = 1

sin (x) = 1

sin (x) = 1

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}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

因为方程对以下值无定义:

\frac{π}{2} + 2 πn

x \in R 无解

作图

流行的例子

2cos^2(x)-cos(x)-1=0,(0,2pi)

11=12+3.5sin((2pi)/(365)t)