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受欢迎的三角函数 >

(sec(x)+csc(x))/(1+tan(x))=sin(x)

解答

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} = sin (x)

解答

x \in R 无解

求解步骤

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} = sin (x)

两边减去

sin (x)

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x) = 0

化简

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x) : \frac{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )}

\frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - sin (x)

将项转换为分式:

sin (x) = \frac{sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )}

= \frac{sec ( x ) + csc ( x )}{1 + tan ( x )} - \frac{sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( 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{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )}

\frac{sec ( x ) + csc ( x ) - sin ( x ) ( 1 + tan ( x ) )}{1 + tan ( x )} = 0

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

sec (x) + csc (x) - sin (x) (1 + tan (x)) = 0

用 sin, cos 表示

csc (x) + sec (x) - (1 + tan (x)) sin (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

\frac{1}{sin ( x )} + \frac{1}{cos ( x )} - (1 + \frac{sin ( x )}{cos ( x )}) sin (x) : \frac{cos ( x ) + sin ( x ) - sin ^{2} ( x ) ( cos ( x ) + sin ( x ) )}{sin ( x ) cos ( x )}

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

化简

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

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

将项转换为分式:

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

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

乘以:

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

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

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

乘

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

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

分式相乘:

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

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

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

sin (x), cos (x), cos (x)

的最小公倍数:

sin (x) cos (x)

sin (x), cos (x), cos (x)

最小公倍数 (LCM)

计算出由至少在以下一个因式表达式中出现的因子组成的表达式

= sin (x) cos (x)

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

sin (x) cos (x)

对于

\frac{1}{sin ( x )} :

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

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

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

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

分解

cos (x) + sin (x) - (cos (x) + sin (x)) sin^{2} (x) : - (cos (x) + sin (x)) (sin (x) + 1) (sin (x) - 1)

cos (x) + sin (x) - (cos (x) + sin (x)) sin^{2} (x)

改写为

= - (cos (x) + sin (x)) sin^{2} (x) + 1 \cdot (cos (x) + sin (x))

因式分解出通项

(cos (x) + sin (x))

= (cos (x) + sin (x)) (- sin^{2} (x) + 1)

分解

- sin^{2} (x) + 1 : - (sin (x) + 1) (sin (x) - 1)

- sin^{2} (x) + 1

因式分解出通项

- 1

= - (sin^{2} (x) - 1)

分解

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

将

1

改写为

1^{2}

= sin^{2} (x) - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - 1^{2} = (sin (x) + 1) (sin (x) - 1)

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

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

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

整理后得

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

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

分别求解每个部分

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

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) + sin (x) = 0

使用三角恒等式改写

cos (x) + sin (x) = 0

在两边除以

cos (x), cos (x) \neq = 0

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

化简

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

使用基本三角恒等式:

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

1 + tan (x) = 0

1 + tan (x) = 0

将

1

到右边

1 + tan (x) = 0

两边减去

1

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

化简

tan (x) = - 1

tan (x) = - 1

tan (x) = - 1

的通解

tan (x)

周期表（周期为

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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

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

sin (x) - 1 = 0

将

1

到右边

sin (x) - 1 = 0

两边加上

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{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

合并所有解

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

因为方程对以下值无定义:

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

x \in R 无解

作图

流行的例子

sec(x)-(sin(x))/(cos(x))=cos(x)

cos(θ)=0.4,sec(θ)

cot(θ)=(1+cos^2(θ))/(2sin(θ)cos(θ))

sec(3x)-csc(30)=0,(x+35)/5

4cos(2θ)-10cos(θ)+14=7,0<= θ<360