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受欢迎的三角函数 >

8tan(x/2)+8cos(x)tan(x/2)=1

解答

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) = 1

解答

x = 0.12532 \dots + 2 πn, x = π - 0.12532 \dots + 2 πn

+1

度数

x = 7.18075 \dots^{\circ} + 36 0^{\circ} n, x = 172.81924 \dots^{\circ} + 36 0^{\circ} n

求解步骤

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) = 1

两边减去

1

8 tan (\frac{x}{2}) + 8 cos (x) tan (\frac{x}{2}) - 1 = 0

令

: u = \frac{x}{2}

8 tan (u) + 8 cos (2 u) tan (u) - 1 = 0

使用三角恒等式改写

- 1 + 8 tan (u) + 8 cos (2 u) tan (u)

使用倍角公式:

cos (2 x) = 2 cos^{2} (x) - 1

= - 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

化简

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 8 (2 cos^{2} (u) - 1) tan (u)

= - 1 + 8 tan (u) + 8 tan (u) (2 cos^{2} (u) - 1)

乘开

8 tan (u) (2 cos^{2} (u) - 1) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 tan (u) (2 cos^{2} (u) - 1)

使用分配律:

a (b - c) = ab - a c

a = 8 tan (u), b = 2 cos^{2} (u), c = 1

= 8 tan (u) \cdot 2 cos^{2} (u) - 8 tan (u) \cdot 1

= 8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

化简

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u) : 16 cos^{2} (u) tan (u) - 8 tan (u)

8 \cdot 2 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

数字相乘:

8 \cdot 2 = 16

= 16 cos^{2} (u) tan (u) - 8 \cdot 1 \cdot tan (u)

数字相乘:

8 \cdot 1 = 8

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= 16 cos^{2} (u) tan (u) - 8 tan (u)

= - 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

化简

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) : 16 cos^{2} (u) tan (u) - 1

- 1 + 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u)

对同类项分组

= 8 tan (u) + 16 cos^{2} (u) tan (u) - 8 tan (u) - 1

同类项相加:

8 tan (u) - 8 tan (u) = 0

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

= 16 cos^{2} (u) tan (u) - 1

使用基本三角恒等式:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 1 + 16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )} = 16 sin (u) cos (u)

16 cos^{2} (u) \frac{sin ( u )}{cos ( u )}

分式相乘:

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

= \frac{sin ( u ) \cdot 16 cos ^{2} ( u )}{cos ( u )}

约分:

cos (u)

= 16 sin (u) cos (u)

= - 1 + 16 sin (u) cos (u)

使用倍角公式:

2 sin (x) cos (x) = sin (2 x)

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

= - 1 + 16 \cdot \frac{sin ( 2 u )}{2}

- 1 + 16 \cdot \frac{sin ( 2 u )}{2} = 0

16 \cdot \frac{sin ( 2 u )}{2} = 8 sin (2 u)

16 \cdot \frac{sin ( 2 u )}{2}

分式相乘:

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

= \frac{sin ( 2 u ) \cdot 16}{2}

数字相除:

\frac{16}{2} = 8

= 8 sin (2 u)

- 1 + 8 sin (2 u) = 0

将

1

到右边

- 1 + 8 sin (2 u) = 0

两边加上

1

- 1 + 8 sin (2 u) + 1 = 0 + 1

化简

8 sin (2 u) = 1

8 sin (2 u) = 1

两边除以

8

8 sin (2 u) = 1

两边除以

8

\frac{8 sin ( 2 u )}{8} = \frac{1}{8}

化简

sin (2 u) = \frac{1}{8}

sin (2 u) = \frac{1}{8}

使用反三角函数性质

sin (2 u) = \frac{1}{8}

sin (2 u) = \frac{1}{8}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 u = arcsin (\frac{1}{8}) + 2 πn, 2 u = π - arcsin (\frac{1}{8}) + 2 πn

2 u = arcsin (\frac{1}{8}) + 2 πn, 2 u = π - arcsin (\frac{1}{8}) + 2 πn

解

2 u = arcsin (\frac{1}{8}) + 2 πn : u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

2 u = arcsin (\frac{1}{8}) + 2 πn

两边除以

2

2 u = arcsin (\frac{1}{8}) + 2 πn

两边除以

2

\frac{2 u}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + \frac{2 πn}{2}

化简

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn

解

2 u = π - arcsin (\frac{1}{8}) + 2 πn : u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

2 u = π - arcsin (\frac{1}{8}) + 2 πn

两边除以

2

2 u = π - arcsin (\frac{1}{8}) + 2 πn

两边除以

2

\frac{2 u}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + \frac{2 πn}{2}

化简

u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{arcsin ( \frac{1}{8} )}{2} + πn, u = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

u = \frac{x}{2}

代回

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn : x = arcsin (\frac{1}{8}) + 2 πn

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn

在两边乘以

2

\frac{x}{2} = \frac{arcsin ( \frac{1}{8} )}{2} + πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

化简

\frac{2 x}{2} = 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn : arcsin (\frac{1}{8}) + 2 πn

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} = arcsin (\frac{1}{8})

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2}

分式相乘:

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

= \frac{arcsin ( \frac{1}{8} ) \cdot 2}{2}

约分:

2

= arcsin (\frac{1}{8})

= arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn

\frac{x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn : x = π - arcsin (\frac{1}{8}) + 2 πn

\frac{x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

在两边乘以

2

\frac{x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{1}{8} )}{2} + πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

化简

\frac{2 x}{2} = 2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 2 πn : π - arcsin (\frac{1}{8}) + 2 πn

2 \cdot \frac{π}{2} - 2 \cdot \frac{arcsin ( \frac{1}{8} )}{2} + 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 \frac{arcsin ( \frac{1}{8} )}{2} = arcsin (\frac{1}{8})

2 \cdot \frac{arcsin ( \frac{1}{8} )}{2}

分式相乘:

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

= \frac{arcsin ( \frac{1}{8} ) \cdot 2}{2}

约分:

2

= arcsin (\frac{1}{8})

= π - arcsin (\frac{1}{8}) + 2 πn

x = π - arcsin (\frac{1}{8}) + 2 πn

x = π - arcsin (\frac{1}{8}) + 2 πn

x = π - arcsin (\frac{1}{8}) + 2 πn

x = arcsin (\frac{1}{8}) + 2 πn, x = π - arcsin (\frac{1}{8}) + 2 πn

以小数形式表示解

x = 0.12532 \dots + 2 πn, x = π - 0.12532 \dots + 2 πn

作图

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