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受欢迎的三角函数 >

sec(θ/2)=cos(θ/2)

解答

sec (\frac{θ}{2}) = cos (\frac{θ}{2})

解答

θ = 4 πn, θ = 2 π + 4 πn

+1

度数

θ = 0^{\circ} + 72 0^{\circ} n, θ = 36 0^{\circ} + 72 0^{\circ} n

求解步骤

sec (\frac{θ}{2}) = cos (\frac{θ}{2})

两边减去

cos (\frac{θ}{2})

sec (\frac{θ}{2}) - cos (\frac{θ}{2}) = 0

使用三角恒等式改写

- cos (\frac{θ}{2}) + sec (\frac{θ}{2})

使用基本三角恒等式:

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

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

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

用替代法求解

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

令

: sec (\frac{θ}{2}) = u

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

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

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

在两边乘以

u

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

在两边乘以

u

- \frac{1}{u} u + uu = 0 \cdot u

化简

- \frac{1}{u} u + uu = 0 \cdot u

化简

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

- \frac{1}{u} u

分式相乘:

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

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

约分:

u

= - 1

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

- 1 + u^{2} = 0

- 1 + u^{2} = 0

- 1 + u^{2} = 0

解

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

- 1 + u^{2} = 0

将

1

到右边

- 1 + u^{2} = 0

两边加上

1

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

化简

u^{2} = 1

u^{2} = 1

对于

x^{2} = f (a)

解为

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

u = 1, u = - 1

1 = 1

1

使用法则

1 = 1

= 1

- 1 = - 1

- 1

使用法则

1 = 1

= - 1

u = 1, u = - 1

u = 1, u = - 1

验证解

找到无定义的点（奇点）:

u = 0

取

- \frac{1}{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 = 1, u = - 1

u = sec (\frac{θ}{2})

代回

sec (\frac{θ}{2}) = 1, sec (\frac{θ}{2}) = - 1

sec (\frac{θ}{2}) = 1, sec (\frac{θ}{2}) = - 1

sec (\frac{θ}{2}) = 1 : θ = 4 πn

sec (\frac{θ}{2}) = 1

sec (\frac{θ}{2}) = 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}

\frac{θ}{2} = 0 + 2 πn

\frac{θ}{2} = 0 + 2 πn

解

\frac{θ}{2} = 0 + 2 πn : θ = 4 πn

\frac{θ}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{θ}{2} = 2 πn

在两边乘以

2

\frac{θ}{2} = 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot 2 πn

化简

θ = 4 πn

θ = 4 πn

θ = 4 πn

sec (\frac{θ}{2}) = - 1 : θ = 2 π + 4 πn

sec (\frac{θ}{2}) = - 1

sec (\frac{θ}{2}) = - 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}

\frac{θ}{2} = π + 2 πn

\frac{θ}{2} = π + 2 πn

解

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

\frac{θ}{2} = π + 2 πn

在两边乘以

2

\frac{θ}{2} = π + 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 π + 2 \cdot 2 πn

化简

θ = 2 π + 4 πn

θ = 2 π + 4 πn

θ = 2 π + 4 πn

合并所有解

θ = 4 πn, θ = 2 π + 4 πn

作图

流行的例子

2cos^2(x)-1.28=0

2 cos^{2} (x) - 1.28 = 0

cos^3(3θ)= 1/4

cos^{3} (3 θ) = \frac{1}{4}

sin(x-pi/4)= 1/2

sin (x - \frac{π}{4}) = \frac{1}{2}

3sin(2x)-3/2 sqrt(3)=0

3 sin (2 x) - \frac{3}{2} 3 = 0

sin^2(θ)-1/4 =0

sin^{2} (θ) - \frac{1}{4} = 0