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受欢迎的三角函数 >

((sin(x)+tan(x)))/(1+cos(x))=m*sec(x)

解答

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

解答

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

求解步骤

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

两边减去

m sec (x)

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

化简

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

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

将项转换为分式:

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

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

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

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

sin (x) + tan (x) - m sec (x) (1 + cos (x)) = 0

用 sin, cos 表示

sin (x) + tan (x) - (1 + cos (x)) sec (x) m

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

乘以:

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

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

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

合并分式

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

使用法则

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

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

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

将项转换为分式:

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

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

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

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

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

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

分解

sin (x) - (1 + cos (x)) m + cos (x) sin (x) : (1 + cos (x)) (sin (x) - m)

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

因式分解出通项

sin (x)

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

因式分解出通项

(1 + cos (x))

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

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

分别求解每个部分

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

1 + cos (x) = 0 : x = π + 2 πn

1 + cos (x) = 0

将

1

到右边

1 + cos (x) = 0

两边减去

1

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

化简

cos (x) = - 1

cos (x) = - 1

cos (x) = - 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

sin (x) - m = 0 : x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

sin (x) - m = 0

将

m

到右边

sin (x) - m = 0

两边加上

m

sin (x) - m + m = 0 + m

化简

sin (x) = m

sin (x) = m

使用反三角函数性质

sin (x) = m

sin (x) = m

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

合并所有解

x = π + 2 πn, x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

因为方程对以下值无定义:

π + 2 πn

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

作图

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