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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{4} + πn

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

1

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

化简

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

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

将项转换为分式:

1 = \frac{1}{1}

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

csc (x), sec (x), 1

的最小公倍数:

csc (x) sec (x)

csc (x), sec (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

= csc (x) sec (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

csc (x) sec (x)

对于

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

将分母和分子乘以

sec (x)

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

对于

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

将分母和分子乘以

csc (x)

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

对于

\frac{1}{1} :

将分母和分子乘以

csc (x) sec (x)

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

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

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

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

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

用 sin, cos 表示

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

使用基本三角恒等式:

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

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

使用基本三角恒等式:

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

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

化简

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

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

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

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

分式相乘:

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

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

约分:

cos (x)

= 1

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

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

分式相乘:

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

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

约分:

sin (x)

= 1

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

对同类项分组

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

数字相加:

1 + 1 = 2

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

将项转换为分式:

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

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

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

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

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

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

使用三角恒等式改写

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

使用倍角公式:

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

= - 1 + sin (2 x)

- 1 + sin (2 x) = 0

将

1

到右边

- 1 + sin (2 x) = 0

两边加上

1

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

化简

sin (2 x) = 1

sin (2 x) = 1

sin (2 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}

2 x = \frac{π}{2} + 2 πn

2 x = \frac{π}{2} + 2 πn

解

2 x = \frac{π}{2} + 2 πn : x = \frac{π}{4} + πn

2 x = \frac{π}{2} + 2 πn

两边除以

2

2 x = \frac{π}{2} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2} : \frac{π}{4} + πn

\frac{\frac{π}{2}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

使用分式法则:

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

= \frac{π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

作图

流行的例子

cot (θ) = - 8

solvefor t,0=(1.8)cos(2pift)

so l v e f or t, 0 = (1.8) cos (2 π f t)

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

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

tan(θ)-7=-12cot(θ)

tan (θ) - 7 = - 12 cot (θ)

4 cos^{2} (3 x) - 1 = 0