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受欢迎的三角函数 >

cos(x)=sec(x)(1-cos^2(x))

解答

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

解答

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

+1

度数

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

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

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

使用三角恒等式改写

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

使用基本三角恒等式:

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

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

化简

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

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = - sec (x), b = 1, c = \frac{1}{sec ^{2} ( x )}

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

使用加减运算法则

- (- a) = a

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

化简

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

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

1 \cdot sec (x) = sec (x)

1 \cdot sec (x)

乘以:

1 \cdot sec (x) = sec (x)

= sec (x)

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

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

分式相乘:

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

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

乘以:

1 \cdot sec (x) = sec (x)

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

约分:

sec (x)

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

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

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

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

合并分式

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

使用法则

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

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

数字相加:

1 + 1 = 2

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

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

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

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

用替代法求解

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

令

: sec (x) = u

\frac{2}{u} - u = 0

\frac{2}{u} - u = 0 : u = 2, u = - 2

\frac{2}{u} - u = 0

在两边乘以

u

\frac{2}{u} - u = 0

在两边乘以

u

\frac{2}{u} u - uu = 0 \cdot u

化简

\frac{2}{u} u - uu = 0 \cdot u

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

化简

- uu : - u^{2}

- uu

使用指数法则:

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

uu = u^{1 + 1}

= - u^{1 + 1}

数字相加:

1 + 1 = 2

= - u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

2 - u^{2} = 0

2 - u^{2} = 0

2 - u^{2} = 0

解

2 - u^{2} = 0 : u = 2, u = - 2

2 - u^{2} = 0

将

2

到右边

2 - u^{2} = 0

两边减去

2

2 - u^{2} - 2 = 0 - 2

化简

- u^{2} = - 2

- u^{2} = - 2

两边除以

- 1

- u^{2} = - 2

两边除以

- 1

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

化简

u^{2} = 2

u^{2} = 2

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = 2, u = - 2

u = 2, u = - 2

验证解

找到无定义的点（奇点）:

u = 0

取

\frac{2}{u} - u

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = 2, u = - 2

u = sec (x)

代回

sec (x) = 2, sec (x) = - 2

sec (x) = 2, sec (x) = - 2

sec (x) = 2 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = 2

sec (x) = 2

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 = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) = - 2 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) = - 2

sec (x) = - 2

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 = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

合并所有解

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

作图

流行的例子

sin (t) = \frac{2}{3}

1/485 tan^2(x)=0

\frac{1}{485} tan^{2} (x) = 0

5 cos (3 x) = - 4

sin (2 t) = \frac{7}{9}

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

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