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受欢迎的三角函数 >

1/(cos(x))+sec^2(x)-1-cos(x)=0

解答

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

解答

x = π + 2 πn, x = 2 πn

+1

度数

x = 18 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n

求解步骤

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

化简

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

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

将项转换为分式:

sec^{2} (x) = \frac{sec ^{2} ( x ) cos ( x )}{cos ( x )}, 1 = \frac{1 cos ( x )}{cos ( x )}, cos (x) = \frac{cos ( x ) cos ( x )}{cos ( x )}

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

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

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

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

1 \cdot cos (x)

乘以:

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

= cos (x)

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

cos (x) cos (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

数字相加:

1 + 1 = 2

= cos^{2} (x)

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

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

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

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

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

使用三角恒等式改写

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

使用基本三角恒等式:

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

= 1 - \frac{1}{sec ( x )} - (\frac{1}{sec ( x )})^{2} + \frac{1}{sec ( x )} sec^{2} (x)

化简

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

1 - \frac{1}{sec ( x )} - (\frac{1}{sec ( x )})^{2} + \frac{1}{sec ( x )} sec^{2} (x)

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sec ^{2} ( x )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

\frac{1}{sec ( x )} sec^{2} (x) = sec (x)

\frac{1}{sec ( x )} sec^{2} (x)

分式相乘:

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

= \frac{1 \cdot sec ^{2} ( x )}{sec ( x )}

乘以:

1 \cdot sec^{2} (x) = sec^{2} (x)

= \frac{sec ^{2} ( x )}{sec ( x )}

约分:

sec (x)

= sec (x)

= 1 - \frac{1}{sec ( x )} - \frac{1}{sec ^{2} ( x )} + sec (x)

= 1 - \frac{1}{sec ( x )} - \frac{1}{sec ^{2} ( x )} + sec (x)

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

用替代法求解

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

令

: sec (x) = u

1 - \frac{1}{u ^{2}} - \frac{1}{u} + u = 0

1 - \frac{1}{u ^{2}} - \frac{1}{u} + u = 0 : u = - 1, u = 1

1 - \frac{1}{u ^{2}} - \frac{1}{u} + u = 0

乘以最小公倍数

1 - \frac{1}{u ^{2}} - \frac{1}{u} + u = 0

找到

u^{2}, u

的最小公倍数:

u^{2}

u^{2}, u

最小公倍数 (LCM)

计算出由出现在

u^{2}

或

u

中的因子组成的表达式

= u^{2}

乘以最小公倍数=

u^{2}

1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} - \frac{1}{u} u^{2} + u u^{2} = 0 \cdot u^{2}

化简

1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} - \frac{1}{u} u^{2} + u u^{2} = 0 \cdot u^{2}

化简

1 \cdot u^{2} : u^{2}

1 \cdot u^{2}

乘以:

1 \cdot u^{2} = u^{2}

= u^{2}

化简

- \frac{1}{u ^{2}} u^{2} : - 1

- \frac{1}{u ^{2}} u^{2}

分式相乘:

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

= - \frac{1 \cdot u ^{2}}{u ^{2}}

约分:

u^{2}

= - 1

化简

- \frac{1}{u} u^{2} : - u

- \frac{1}{u} u^{2}

分式相乘:

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

= - \frac{1 \cdot u ^{2}}{u}

乘以:

1 \cdot u^{2} = u^{2}

= - \frac{u ^{2}}{u}

约分:

u

= - u

化简

u u^{2} : u^{3}

u u^{2}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

u u^{2} = u^{1 + 2}

= u^{1 + 2}

数字相加:

1 + 2 = 3

= u^{3}

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

u^{2} - 1 - u + u^{3} = 0

u^{2} - 1 - u + u^{3} = 0

u^{2} - 1 - u + u^{3} = 0

解

u^{2} - 1 - u + u^{3} = 0 : u = - 1, u = 1

u^{2} - 1 - u + u^{3} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{3} + u^{2} - u - 1 = 0

因式分解

u^{3} + u^{2} - u - 1 : (u + 1)^{2} (u - 1)

u^{3} + u^{2} - u - 1

= (u^{3} + u^{2}) + (- u - 1)

从

- u - 1

分解出因式

- 1

:

- (u + 1)

- u - 1

因式分解出通项

- 1

= - (u + 1)

从

u^{3} + u^{2}

分解出因式

u^{2}

:

u^{2} (u + 1)

u^{3} + u^{2}

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{3} = u u^{2}

= u u^{2} + u^{2}

因式分解出通项

u^{2}

= u^{2} (u + 1)

= - (u + 1) + u^{2} (u + 1)

因式分解出通项

u + 1

= (u + 1) (u^{2} - 1)

分解

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

将

1

改写为

1^{2}

= u^{2} - 1^{2}

使用平方差公式:

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

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= (u + 1) (u + 1) (u - 1)

整理后得

= (u + 1)^{2} (u - 1)

(u + 1)^{2} (u - 1) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u + 1 = 0 or u - 1 = 0

解

u + 1 = 0 : u = - 1

u + 1 = 0

将

1

到右边

u + 1 = 0

两边减去

1

u + 1 - 1 = 0 - 1

化简

u = - 1

u = - 1

解

u - 1 = 0 : u = 1

u - 1 = 0

将

1

到右边

u - 1 = 0

两边加上

1

u - 1 + 1 = 0 + 1

化简

u = 1

u = 1

解为

u = - 1, u = 1

u = - 1, u = 1

验证解

找到无定义的点（奇点）:

u = 0

取

1 - \frac{1}{u ^{2}} - \frac{1}{u} + u

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = - 1, u = 1

u = sec (x)

代回

sec (x) = - 1, sec (x) = 1

sec (x) = - 1, sec (x) = 1

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

sec (x) = - 1

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

sec (x) = 1 : x = 2 πn

sec (x) = 1

sec (x) = 1

的通解

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

合并所有解

x = π + 2 πn, x = 2 πn

作图

流行的例子

tan(θ)=(sqrt(3))/3 ,180<= θ<= 360

tan (θ) = \frac{3}{3}, 18 0^{\circ} \leq θ \leq 36 0^{\circ}

7.5=5sin(pi/2 (x-2))+5

7.5 = 5 sin (\frac{π}{2} (x - 2)) + 5

2cos(θ)+2sqrt(3)=sqrt(3)

2 cos (θ) + 23 = 3

-2sqrt(2)=-4sin(2θ)

- 22 = - 4 sin (2 θ)

8cos(x)=4cos^2(x)+3

8 cos (x) = 4 cos^{2} (x) + 3