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csc(θ/2)=sin(θ/2)

解答

csc (\frac{θ}{2}) = sin (\frac{θ}{2})

解答

θ = π + 4 πn, θ = 3 π + 4 πn

+1

度数

θ = 18 0^{\circ} + 72 0^{\circ} n, θ = 54 0^{\circ} + 72 0^{\circ} n

求解步骤

csc (\frac{θ}{2}) = sin (\frac{θ}{2})

两边减去

sin (\frac{θ}{2})

csc (\frac{θ}{2}) - sin (\frac{θ}{2}) = 0

使用三角恒等式改写

csc (\frac{θ}{2}) - sin (\frac{θ}{2})

使用基本三角恒等式:

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

= csc (\frac{θ}{2}) - \frac{1}{csc ( \frac{θ}{2} )}

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

用替代法求解

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

令

: csc (\frac{θ}{2}) = u

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

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

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

在两边乘以

u

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

在两边乘以

u

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

化简

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

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

- \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

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

u^{2} - 1 = 0

u^{2} - 1 = 0

u^{2} - 1 = 0

解

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

u^{2} - 1 = 0

将

1

到右边

u^{2} - 1 = 0

两边加上

1

u^{2} - 1 + 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

u - \frac{1}{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 = csc (\frac{θ}{2})

代回

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

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

csc (\frac{θ}{2}) = 1 : θ = π + 4 πn

csc (\frac{θ}{2}) = 1

csc (\frac{θ}{2}) = 1

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

\frac{θ}{2} = \frac{π}{2} + 2 πn

\frac{θ}{2} = \frac{π}{2} + 2 πn

解

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

\frac{θ}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{θ}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分式相乘:

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

= \frac{π 2}{2}

约分:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

θ = π + 4 πn

θ = π + 4 πn

θ = π + 4 πn

θ = π + 4 πn

csc (\frac{θ}{2}) = - 1 : θ = 3 π + 4 πn

csc (\frac{θ}{2}) = - 1

csc (\frac{θ}{2}) = - 1

的通解

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

解

\frac{θ}{2} = \frac{3 π}{2} + 2 πn : θ = 3 π + 4 πn

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{θ}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

分式相乘:

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

= \frac{3 π 2}{2}

约分:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

θ = 3 π + 4 πn

θ = 3 π + 4 πn

θ = 3 π + 4 πn

θ = 3 π + 4 πn

合并所有解

θ = π + 4 πn, θ = 3 π + 4 πn

流行的例子

sin(3x)=-1,0<= x<= 2pi

sin (3 x) = - 1, 0 \leq x \leq 2 π

sin (θ) = 0.25

30^2=20^2+18^2-2(20)(18)cos(A)

3 0^{2} = 2 0^{2} + 1 8^{2} - 2 (20) (18) cos (A)

cos(x)= 308/1475

cos (x) = \frac{308}{1475}

tan (θ) = \frac{6}{3}