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受欢迎的三角函数 >

(cos(x))/(sec(x))-(sin(x))/(csc(x))=1

解答

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

解答

x \in R 无解

求解步骤

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

两边减去

1

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

化简

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

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

将项转换为分式:

1 = \frac{1}{1}

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

sec (x), csc (x), 1

的最小公倍数:

sec (x) csc (x)

sec (x), csc (x), 1

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

= sec (x) csc (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

sec (x) csc (x)

对于

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

将分母和分子乘以

csc (x)

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

对于

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

将分母和分子乘以

sec (x)

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

对于

\frac{1}{1} :

将分母和分子乘以

sec (x) csc (x)

\frac{1}{1} = \frac{1 \cdot sec ( x ) csc ( x )}{1 \cdot sec ( x ) csc ( x )} = \frac{sec ( x ) csc ( x )}{sec ( x ) csc ( x )}

= \frac{cos ( x ) csc ( x )}{sec ( x ) csc ( x )} - \frac{sin ( x ) sec ( x )}{csc ( x ) sec ( x )} - \frac{sec ( x ) csc ( x )}{sec ( x ) csc ( 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 ) csc ( x ) - sin ( x ) sec ( x ) - sec ( x ) csc ( x )}{sec ( x ) csc ( x )}

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

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

cos (x) csc (x) - sin (x) sec (x) - sec (x) csc (x) = 0

用 sin, cos 表示

cos (x) csc (x) - csc (x) sec (x) - sec (x) sin (x)

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

乘以:

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

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

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 1 = 1

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

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

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

分式相乘:

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

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

乘以:

1 \cdot sin (x) = sin (x)

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

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

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

的最小公倍数:

sin (x) cos (x)

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

将分母和分子乘以

cos (x)

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

对于

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

将分母和分子乘以

sin (x)

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

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

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

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

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

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

使用三角恒等式改写

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

使用倍角公式:

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

= - 1 + cos (2 x)

- 1 + cos (2 x) = 0

将

1

到右边

- 1 + cos (2 x) = 0

两边加上

1

- 1 + cos (2 x) + 1 = 0 + 1

化简

cos (2 x) = 1

cos (2 x) = 1

cos (2 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}

2 x = 0 + 2 πn

2 x = 0 + 2 πn

解

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

两边除以

2

2 x = 2 πn

两边除以

2

\frac{2 x}{2} = \frac{2 πn}{2}

化简

x = πn

x = πn

x = πn

因为方程对以下值无定义:

πn

x \in R 无解

作图

流行的例子

tan (x) = \frac{12}{35}

4 = sec (θ)

1+sqrt(2)sin(x)=cos(2x)

1 + 2 sin (x) = cos (2 x)

12-sin(θ)=cos(2θ)

12 - sin (θ) = cos (2 θ)

2 cos (θ) = 3