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sec(x)=2cos(x)

解答

sec (x) = 2 cos (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

sec (x) = 2 cos (x)

两边减去

2 cos (x)

sec (x) - 2 cos (x) = 0

使用三角恒等式改写

sec (x) - 2 cos (x)

使用基本三角恒等式:

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

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

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 2 = 2

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

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

两边加上

2

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

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

流行的例子

sec(5 θ/4)=2,0<θ<2pi

sec (5 \frac{θ}{4}) = 2, 0 < θ < 2 π

2 sin (\frac{x}{2}) = 1

sin^2(x)-3cos^2(x)=0

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

cos(2x+pi/3)=sin(3x)

cos (2 x + \frac{π}{3}) = sin (3 x)

solvefor y,x=arcsin(y)求解

y, x = arcsin (y)