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受欢迎的三角函数 >

tan(x)+sec(x)=0

解答

tan (x) + sec (x) = 0

解答

x \in R 无解

求解步骤

tan (x) + sec (x) = 0

使用三角恒等式改写

sec (x) + tan (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

= \frac{sin ( x )}{\frac{1}{s e c ( x )}} + sec (x)

\frac{sin ( x )}{\frac{1}{s e c ( x )}} = sin (x) sec (x)

\frac{sin ( x )}{\frac{1}{s e c ( x )}}

使用分式法则:

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

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

使用法则

\frac{a}{1} = a

= sin (x) sec (x)

= sin (x) sec (x) + sec (x)

sec (x) + sec (x) sin (x) = 0

分解

sec (x) + sec (x) sin (x) : sec (x) (sin (x) + 1)

sec (x) + sec (x) sin (x)

因式分解出通项

sec (x)

= sec (x) (1 + sin (x))

sec (x) (sin (x) + 1) = 0

分别求解每个部分

sec (x) = 0 or sin (x) + 1 = 0

sec (x) = 0 :

无解

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

无解

sin (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

sin (x) + 1 = 0

将

1

到右边

sin (x) + 1 = 0

两边减去

1

sin (x) + 1 - 1 = 0 - 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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

x = \frac{3 π}{2} + 2 πn

因为方程对以下值无定义:

\frac{3 π}{2} + 2 πn

x \in R 无解

作图

流行的例子

cos(θ)=sqrt(3/2)

cos (θ) = \frac{3}{2}

cos^2(θ)+2sin(θ)cos(θ)-sin^2(θ)=0

cos^{2} (θ) + 2 sin (θ) cos (θ) - sin^{2} (θ) = 0

tan(x)+tan^2(x)=0

tan (x) + tan^{2} (x) = 0

sin(2θ)+2sin(θ)=0

sin (2 θ) + 2 sin (θ) = 0

sin (x + \frac{π}{2}) = 1