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受欢迎的三角函数 >

2sin^2(θ/2-pi/6)-1=0

解答

2 sin^{2} (\frac{θ}{2} - \frac{π}{6}) - 1 = 0

解答

θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}, θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

+1

度数

θ = 15 0^{\circ} + 72 0^{\circ} n, θ = 33 0^{\circ} + 72 0^{\circ} n, θ = 51 0^{\circ} + 72 0^{\circ} n, θ = 69 0^{\circ} + 72 0^{\circ} n

求解步骤

2 sin^{2} (\frac{θ}{2} - \frac{π}{6}) - 1 = 0

用替代法求解

2 sin^{2} (\frac{θ}{2} - \frac{π}{6}) - 1 = 0

令

: sin (\frac{θ}{2} - \frac{π}{6}) = u

2 u^{2} - 1 = 0

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

2 u^{2} - 1 = 0

将

1

到右边

2 u^{2} - 1 = 0

两边加上

1

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

化简

2 u^{2} = 1

2 u^{2} = 1

两边除以

2

2 u^{2} = 1

两边除以

2

\frac{2 u ^{2}}{2} = \frac{1}{2}

化简

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2}

对于

x^{2} = f (a)

解为

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

u = \frac{1}{2}, u = - \frac{1}{2}

u = sin (\frac{θ}{2} - \frac{π}{6})

代回

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2}, sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2}

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2}, sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2}

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2} : θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2}

sin (\frac{θ}{2} - \frac{π}{6}) = \frac{1}{2}

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

解

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn : θ = 4 πn + \frac{5 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn

将

\frac{π}{6}

到右边

\frac{θ}{2} - \frac{π}{6} = \frac{π}{4} + 2 πn

两边加上

\frac{π}{6}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

同类项相加:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

化简

\frac{π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{5 π}{12}

\frac{π}{4} + 2 πn + \frac{π}{6}

对同类项分组

= 2 πn + \frac{π}{4} + \frac{π}{6}

4, 6

的最小公倍数:

12

4, 6

最小公倍数 (LCM)

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

将每个因子乘以它在

4

或

6

中出现的最多次数

= 2 \cdot 2 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 = 12

= 12

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

12

对于

\frac{π}{4} :

将分母和分子乘以

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

对于

\frac{π}{6} :

将分母和分子乘以

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

= \frac{π 3}{12} + \frac{π 2}{12}

因为分母相等，所以合并分式:

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

= \frac{π 3 + π 2}{12}

同类项相加:

3 π + 2 π = 5 π

= 2 πn + \frac{5 π}{12}

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

在两边乘以

2

\frac{θ}{2} = 2 πn + \frac{5 π}{12}

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{5 π}{12}

化简

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{5 π}{12}

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

2 \cdot 2 πn + 2 \cdot \frac{5 π}{12} : 4 πn + \frac{5 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{5 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{5 π}{12} = \frac{5 π}{6}

2 \cdot \frac{5 π}{12}

分式相乘:

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

= \frac{5 π 2}{12}

数字相乘:

5 \cdot 2 = 10

= \frac{10 π}{12}

约分:

2

= \frac{5 π}{6}

= 4 πn + \frac{5 π}{6}

θ = 4 πn + \frac{5 π}{6}

θ = 4 πn + \frac{5 π}{6}

θ = 4 πn + \frac{5 π}{6}

解

\frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn : θ = 4 πn + \frac{11 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

将

\frac{π}{6}

到右边

\frac{θ}{2} - \frac{π}{6} = \frac{3 π}{4} + 2 πn

两边加上

\frac{π}{6}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{3 π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{3 π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

同类项相加:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

化简

\frac{3 π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{11 π}{12}

\frac{3 π}{4} + 2 πn + \frac{π}{6}

对同类项分组

= 2 πn + \frac{π}{6} + \frac{3 π}{4}

6, 4

的最小公倍数:

12

6, 4

最小公倍数 (LCM)

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

将每个因子乘以它在

6

或

4

中出现的最多次数

= 2 \cdot 2 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 = 12

= 12

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

12

对于

\frac{π}{6} :

将分母和分子乘以

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

对于

\frac{3 π}{4} :

将分母和分子乘以

3

\frac{3 π}{4} = \frac{3 π 3}{4 \cdot 3} = \frac{9 π}{12}

= \frac{π 2}{12} + \frac{9 π}{12}

因为分母相等，所以合并分式:

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

= \frac{π 2 + 9 π}{12}

同类项相加:

2 π + 9 π = 11 π

= 2 πn + \frac{11 π}{12}

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

在两边乘以

2

\frac{θ}{2} = 2 πn + \frac{11 π}{12}

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{11 π}{12}

化简

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{11 π}{12}

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

2 \cdot 2 πn + 2 \cdot \frac{11 π}{12} : 4 πn + \frac{11 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{11 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{11 π}{12} = \frac{11 π}{6}

2 \cdot \frac{11 π}{12}

分式相乘:

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

= \frac{11 π 2}{12}

数字相乘:

11 \cdot 2 = 22

= \frac{22 π}{12}

约分:

2

= \frac{11 π}{6}

= 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{11 π}{6}

θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}

sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2} : θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2}

sin (\frac{θ}{2} - \frac{π}{6}) = - \frac{1}{2}

的通解

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn, \frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

解

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn : θ = 4 πn + \frac{17 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn

将

\frac{π}{6}

到右边

\frac{θ}{2} - \frac{π}{6} = \frac{5 π}{4} + 2 πn

两边加上

\frac{π}{6}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{5 π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{5 π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

同类项相加:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

化简

\frac{5 π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{17 π}{12}

\frac{5 π}{4} + 2 πn + \frac{π}{6}

对同类项分组

= 2 πn + \frac{π}{6} + \frac{5 π}{4}

6, 4

的最小公倍数:

12

6, 4

最小公倍数 (LCM)

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

将每个因子乘以它在

6

或

4

中出现的最多次数

= 2 \cdot 2 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 = 12

= 12

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

12

对于

\frac{π}{6} :

将分母和分子乘以

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

对于

\frac{5 π}{4} :

将分母和分子乘以

3

\frac{5 π}{4} = \frac{5 π 3}{4 \cdot 3} = \frac{15 π}{12}

= \frac{π 2}{12} + \frac{15 π}{12}

因为分母相等，所以合并分式:

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

= \frac{π 2 + 15 π}{12}

同类项相加:

2 π + 15 π = 17 π

= 2 πn + \frac{17 π}{12}

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

在两边乘以

2

\frac{θ}{2} = 2 πn + \frac{17 π}{12}

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

化简

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

2 \cdot 2 πn + 2 \cdot \frac{17 π}{12} : 4 πn + \frac{17 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{17 π}{12} = \frac{17 π}{6}

2 \cdot \frac{17 π}{12}

分式相乘:

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

= \frac{17 π 2}{12}

数字相乘:

17 \cdot 2 = 34

= \frac{34 π}{12}

约分:

2

= \frac{17 π}{6}

= 4 πn + \frac{17 π}{6}

θ = 4 πn + \frac{17 π}{6}

θ = 4 πn + \frac{17 π}{6}

θ = 4 πn + \frac{17 π}{6}

解

\frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn : θ = 4 πn + \frac{23 π}{6}

\frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

将

\frac{π}{6}

到右边

\frac{θ}{2} - \frac{π}{6} = \frac{7 π}{4} + 2 πn

两边加上

\frac{π}{6}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{7 π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} = \frac{7 π}{4} + 2 πn + \frac{π}{6}

化简

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6} : \frac{θ}{2}

\frac{θ}{2} - \frac{π}{6} + \frac{π}{6}

同类项相加:

- \frac{π}{6} + \frac{π}{6} = 0

= \frac{θ}{2}

化简

\frac{7 π}{4} + 2 πn + \frac{π}{6} : 2 πn + \frac{23 π}{12}

\frac{7 π}{4} + 2 πn + \frac{π}{6}

对同类项分组

= 2 πn + \frac{π}{6} + \frac{7 π}{4}

6, 4

的最小公倍数:

12

6, 4

最小公倍数 (LCM)

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

将每个因子乘以它在

6

或

4

中出现的最多次数

= 2 \cdot 2 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 = 12

= 12

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

12

对于

\frac{π}{6} :

将分母和分子乘以

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

对于

\frac{7 π}{4} :

将分母和分子乘以

3

\frac{7 π}{4} = \frac{7 π 3}{4 \cdot 3} = \frac{21 π}{12}

= \frac{π 2}{12} + \frac{21 π}{12}

因为分母相等，所以合并分式:

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

= \frac{π 2 + 21 π}{12}

同类项相加:

2 π + 21 π = 23 π

= 2 πn + \frac{23 π}{12}

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

在两边乘以

2

\frac{θ}{2} = 2 πn + \frac{23 π}{12}

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{23 π}{12}

化简

\frac{2 θ}{2} = 2 \cdot 2 πn + 2 \cdot \frac{23 π}{12}

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

2 \cdot 2 πn + 2 \cdot \frac{23 π}{12} : 4 πn + \frac{23 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{23 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{23 π}{12} = \frac{23 π}{6}

2 \cdot \frac{23 π}{12}

分式相乘:

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

= \frac{23 π 2}{12}

数字相乘:

23 \cdot 2 = 46

= \frac{46 π}{12}

约分:

2

= \frac{23 π}{6}

= 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{23 π}{6}

θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

合并所有解

θ = 4 πn + \frac{5 π}{6}, θ = 4 πn + \frac{11 π}{6}, θ = 4 πn + \frac{17 π}{6}, θ = 4 πn + \frac{23 π}{6}

作图

流行的例子

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23cos(x)=-17-cos(x)

23 cos (x) = - 17 - cos (x)

8cos^2(θ)+14cos(θ)-11=8cos(θ)-6

8 cos^{2} (θ) + 14 cos (θ) - 11 = 8 cos (θ) - 6

sin^2(x)(tan^2(x))=1-sin^2(x)

sin^{2} (x) (tan^{2} (x)) = 1 - sin^{2} (x)

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2 sin^{2} (x) = 3 - 5 cos (x)