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受欢迎的三角函数 >

sec(β/4)csc(β/4)=2csc(β/4)

解答

sec (\frac{β}{4}) csc (\frac{β}{4}) = 2 csc (\frac{β}{4})

解答

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

+1

度数

β = 24 0^{\circ} + 144 0^{\circ} n, β = 120 0^{\circ} + 144 0^{\circ} n

求解步骤

sec (\frac{β}{4}) csc (\frac{β}{4}) = 2 csc (\frac{β}{4})

两边减去

2 csc (\frac{β}{4})

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4}) = 0

分解

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4}) : csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2)

sec (\frac{β}{4}) csc (\frac{β}{4}) - 2 csc (\frac{β}{4})

因式分解出通项

csc (\frac{β}{4})

= csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2)

csc (\frac{β}{4}) (sec (\frac{β}{4}) - 2) = 0

分别求解每个部分

csc (\frac{β}{4}) = 0 or sec (\frac{β}{4}) - 2 = 0

csc (\frac{β}{4}) = 0 :

无解

csc (\frac{β}{4}) = 0

csc (x) \leq - 1 or csc (x) \geq 1

无解

sec (\frac{β}{4}) - 2 = 0 : β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

sec (\frac{β}{4}) - 2 = 0

将

2

到右边

sec (\frac{β}{4}) - 2 = 0

两边加上

2

sec (\frac{β}{4}) - 2 + 2 = 0 + 2

化简

sec (\frac{β}{4}) = 2

sec (\frac{β}{4}) = 2

sec (\frac{β}{4}) = 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}

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

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

解

\frac{β}{4} = \frac{π}{3} + 2 πn : β = \frac{4 π}{3} + 8 πn

\frac{β}{4} = \frac{π}{3} + 2 πn

在两边乘以

4

\frac{β}{4} = \frac{π}{3} + 2 πn

在两边乘以

4

\frac{4 β}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

化简

\frac{4 β}{4} = 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

化简

\frac{4 β}{4} : β

\frac{4 β}{4}

数字相除:

\frac{4}{4} = 1

= β

化简

4 \cdot \frac{π}{3} + 4 \cdot 2 πn : \frac{4 π}{3} + 8 πn

4 \cdot \frac{π}{3} + 4 \cdot 2 πn

乘

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

分式相乘:

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

= \frac{π 4}{3}

= \frac{4 π}{3} + 4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn

解

\frac{β}{4} = \frac{5 π}{3} + 2 πn : β = \frac{20 π}{3} + 8 πn

\frac{β}{4} = \frac{5 π}{3} + 2 πn

在两边乘以

4

\frac{β}{4} = \frac{5 π}{3} + 2 πn

在两边乘以

4

\frac{4 β}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

化简

\frac{4 β}{4} = 4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

化简

\frac{4 β}{4} : β

\frac{4 β}{4}

数字相除:

\frac{4}{4} = 1

= β

化简

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn : \frac{20 π}{3} + 8 πn

4 \cdot \frac{5 π}{3} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{3} = \frac{20 π}{3}

4 \cdot \frac{5 π}{3}

分式相乘:

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

= \frac{5 π 4}{3}

数字相乘:

5 \cdot 4 = 20

= \frac{20 π}{3}

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{20 π}{3} + 8 πn

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

合并所有解

β = \frac{4 π}{3} + 8 πn, β = \frac{20 π}{3} + 8 πn

作图

流行的例子

sin (x) = \frac{1}{7}

tan (\frac{θ}{2}) = - 1

tan (\frac{θ}{2}) = 1

4sin^2(x)cot(x)-2cot(x)=0

4 sin^{2} (x) cot (x) - 2 cot (x) = 0

cos(2θ)+sin(θ)=0

cos (2 θ) + sin (θ) = 0