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受欢迎的三角函数 >

tan(x)=(sec(x))/(cos(x))

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

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

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

化简

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

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

将项转换为分式:

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

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

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

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

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

用 sin, cos 表示

- sec (x) + cos (x) tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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)

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

将项转换为分式:

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

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

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

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

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

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

使用三角恒等式改写

- 1 + cos (x) sin (x)

使用倍角公式:

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

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

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

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

将

1

到右边

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

两边加上

1

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

化简

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

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

在两边乘以

2

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

在两边乘以

2

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

化简

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

x \in R 无解

作图

流行的例子

sin (x) = - 0.707

2sin^2(x)+5sin(x)=0

2 sin^{2} (x) + 5 sin (x) = 0

cos(2x)-7cos(x)-3=0

cos (2 x) - 7 cos (x) - 3 = 0

tan^2(45-x/3)=0.58

tan^{2} (4 5^{\circ} - \frac{x}{3}) = 0.58

- 20 csc (x) = 40