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受欢迎的三角函数 >

csc(3θ)=6sin(3θ),0<θ<2pi

解答

csc (3 θ) = 6 sin (3 θ), 0 < θ < 2 π

解答

θ = \frac{0.42053 \dots}{3}, θ = \frac{π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 2 π}{3}, θ = \frac{3 π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 4 π}{3}, θ = \frac{5 π - 0.42053 \dots}{3}, θ = \frac{π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 2 π}{3}, θ = \frac{3 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 4 π}{3}, θ = \frac{5 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 6 π}{3}

+1

度数

θ = 8.03161 \dots^{\circ}, θ = 51.96838 \dots^{\circ}, θ = 128.03161 \dots^{\circ}, θ = 171.96838 \dots^{\circ}, θ = 248.03161 \dots^{\circ}, θ = 291.96838 \dots^{\circ}, θ = 68.03161 \dots^{\circ}, θ = 111.96838 \dots^{\circ}, θ = 188.03161 \dots^{\circ}, θ = 231.96838 \dots^{\circ}, θ = 308.03161 \dots^{\circ}, θ = 351.96838 \dots^{\circ}

求解步骤

csc (3 θ) = 6 sin (3 θ), 0 < θ < 2 π

两边减去

6 sin (3 θ)

csc (3 θ) - 6 sin (3 θ) = 0

使用三角恒等式改写

csc (3 θ) - 6 sin (3 θ)

使用基本三角恒等式:

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

= csc (3 θ) - 6 \cdot \frac{1}{csc ( 3 θ )}

6 \cdot \frac{1}{csc ( 3 θ )} = \frac{6}{csc ( 3 θ )}

6 \cdot \frac{1}{csc ( 3 θ )}

分式相乘:

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

= \frac{1 \cdot 6}{csc ( 3 θ )}

数字相乘:

1 \cdot 6 = 6

= \frac{6}{csc ( 3 θ )}

= csc (3 θ) - \frac{6}{csc ( 3 θ )}

csc (3 θ) - \frac{6}{csc ( 3 θ )} = 0

用替代法求解

csc (3 θ) - \frac{6}{csc ( 3 θ )} = 0

令

: csc (3 θ) = u

u - \frac{6}{u} = 0

u - \frac{6}{u} = 0 : u = 6, u = - 6

u - \frac{6}{u} = 0

在两边乘以

u

u - \frac{6}{u} = 0

在两边乘以

u

uu - \frac{6}{u} u = 0 \cdot u

化简

uu - \frac{6}{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{6}{u} u : - 6

- \frac{6}{u} u

分式相乘:

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

= - \frac{6 u}{u}

约分:

u

= - 6

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

u^{2} - 6 = 0

u^{2} - 6 = 0

u^{2} - 6 = 0

解

u^{2} - 6 = 0 : u = 6, u = - 6

u^{2} - 6 = 0

将

6

到右边

u^{2} - 6 = 0

两边加上

6

u^{2} - 6 + 6 = 0 + 6

化简

u^{2} = 6

u^{2} = 6

对于

x^{2} = f (a)

解为

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

u = 6, u = - 6

u = 6, u = - 6

验证解

找到无定义的点（奇点）:

u = 0

取

u - \frac{6}{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 = 6, u = - 6

u = csc (3 θ)

代回

csc (3 θ) = 6, csc (3 θ) = - 6

csc (3 θ) = 6, csc (3 θ) = - 6

csc (3 θ) = 6, 0 < θ < 2 π : θ = \frac{arccsc ( 6 )}{3}, θ = \frac{π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π - arccsc ( 6 )}{3}

csc (3 θ) = 6, 0 < θ < 2 π

使用反三角函数性质

csc (3 θ) = 6

csc (3 θ) = 6

的通解

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

3 θ = arccsc (6) + 2 πn, 3 θ = π - arccsc (6) + 2 πn

3 θ = arccsc (6) + 2 πn, 3 θ = π - arccsc (6) + 2 πn

解

3 θ = arccsc (6) + 2 πn : θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = arccsc (6) + 2 πn

两边除以

3

3 θ = arccsc (6) + 2 πn

两边除以

3

\frac{3 θ}{3} = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

化简

θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

解

3 θ = π - arccsc (6) + 2 πn : θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = π - arccsc (6) + 2 πn

两边除以

3

3 θ = π - arccsc (6) + 2 πn

两边除以

3

\frac{3 θ}{3} = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

化简

θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}, θ = \frac{π}{3} - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

在

0 < θ < 2 π

范围内的解

θ = \frac{arccsc ( 6 )}{3}, θ = \frac{π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π - arccsc ( 6 )}{3}

csc (3 θ) = - 6, 0 < θ < 2 π : θ = \frac{π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 6 π}{3}

csc (3 θ) = - 6, 0 < θ < 2 π

使用反三角函数性质

csc (3 θ) = - 6

csc (3 θ) = - 6

的通解

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

3 θ = arccsc (- 6) + 2 πn, 3 θ = π + arccsc (6) + 2 πn

3 θ = arccsc (- 6) + 2 πn, 3 θ = π + arccsc (6) + 2 πn

解

3 θ = arccsc (- 6) + 2 πn : θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = arccsc (- 6) + 2 πn

化简

arccsc (- 6) + 2 πn : - arccsc (6) + 2 πn

arccsc (- 6) + 2 πn

利用以下特性:

arccsc (- x) = - arccsc (x)

arccsc (- 6) = - arccsc (6)

= - arccsc (6) + 2 πn

3 θ = - arccsc (6) + 2 πn

两边除以

3

3 θ = - arccsc (6) + 2 πn

两边除以

3

\frac{3 θ}{3} = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

化简

θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

解

3 θ = π + arccsc (6) + 2 πn : θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

3 θ = π + arccsc (6) + 2 πn

两边除以

3

3 θ = π + arccsc (6) + 2 πn

两边除以

3

\frac{3 θ}{3} = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

化简

θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

θ = - \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}, θ = \frac{π}{3} + \frac{arccsc ( 6 )}{3} + \frac{2 πn}{3}

在

0 < θ < 2 π

范围内的解

θ = \frac{π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 6 π}{3}

合并所有解

θ = \frac{arccsc ( 6 )}{3}, θ = \frac{π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π - arccsc ( 6 )}{3}, θ = \frac{arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π - arccsc ( 6 )}{3}, θ = \frac{π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 2 π}{3}, θ = \frac{3 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 4 π}{3}, θ = \frac{5 π + arccsc ( 6 )}{3}, θ = \frac{- arccsc ( 6 ) + 6 π}{3}

以小数形式表示解

θ = \frac{0.42053 \dots}{3}, θ = \frac{π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 2 π}{3}, θ = \frac{3 π - 0.42053 \dots}{3}, θ = \frac{0.42053 \dots + 4 π}{3}, θ = \frac{5 π - 0.42053 \dots}{3}, θ = \frac{π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 2 π}{3}, θ = \frac{3 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 4 π}{3}, θ = \frac{5 π + 0.42053 \dots}{3}, θ = \frac{- 0.42053 \dots + 6 π}{3}

作图

流行的例子

sqrt(3)tan(3θ)-1=0,0<= θ<= 2pi

3 tan (3 θ) - 1 = 0, 0 \leq θ \leq 2 π

(cot(θ)+1)(csc(θ)-1)=0

(cot (θ) + 1) (csc (θ) - 1) = 0

2sin(x)cos(x)=1

2 sin (x) cos (x) = 1

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

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

cos(x)sin(x)=-sin(x)

cos (x) sin (x) = - sin (x)