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受欢迎的三角函数 >

2sin(2(θ+pi/6))=-sqrt(2)

解答

2 sin (2 (θ + \frac{π}{6})) = - 2

解答

θ = πn + \frac{11 π}{24}, θ = πn + \frac{17 π}{24}

+1

度数

θ = 82. 5^{\circ} + 18 0^{\circ} n, θ = 127. 5^{\circ} + 18 0^{\circ} n

求解步骤

2 sin (2 (θ + \frac{π}{6})) = - 2

两边除以

2

2 sin (2 (θ + \frac{π}{6})) = - 2

两边除以

2

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

化简

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

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

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

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

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

解

2 (θ + \frac{π}{6}) = \frac{5 π}{4} + 2 πn : θ = πn + \frac{11 π}{24}

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

两边除以

2

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

两边除以

2

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

化简

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

化简

\frac{2 ( θ + \frac{π}{6} )}{2} : θ + \frac{π}{6}

\frac{2 ( θ + \frac{π}{6} )}{2}

数字相除:

\frac{2}{2} = 1

= θ + \frac{π}{6}

化简

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

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{4}}{2} = \frac{5 π}{8}

\frac{\frac{5 π}{4}}{2}

使用分式法则:

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

= \frac{5 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{5 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{5 π}{8} + πn

θ + \frac{π}{6} = \frac{5 π}{8} + πn

θ + \frac{π}{6} = \frac{5 π}{8} + πn

θ + \frac{π}{6} = \frac{5 π}{8} + πn

将

\frac{π}{6}

到右边

θ + \frac{π}{6} = \frac{5 π}{8} + πn

两边减去

\frac{π}{6}

θ + \frac{π}{6} - \frac{π}{6} = \frac{5 π}{8} + πn - \frac{π}{6}

化简

θ + \frac{π}{6} - \frac{π}{6} = \frac{5 π}{8} + πn - \frac{π}{6}

化简

θ + \frac{π}{6} - \frac{π}{6} : θ

θ + \frac{π}{6} - \frac{π}{6}

同类项相加:

\frac{π}{6} - \frac{π}{6} = 0

= θ

化简

\frac{5 π}{8} + πn - \frac{π}{6} : πn + \frac{11 π}{24}

\frac{5 π}{8} + πn - \frac{π}{6}

对同类项分组

= πn - \frac{π}{6} + \frac{5 π}{8}

6, 8

的最小公倍数:

24

6, 8

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

8

质因数分解:

2 \cdot 2 \cdot 2

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

将每个因子乘以它在

6

或

8

中出现的最多次数

= 2 \cdot 2 \cdot 2 \cdot 3

数字相乘:

2 \cdot 2 \cdot 2 \cdot 3 = 24

= 24

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

24

对于

\frac{π}{6} :

将分母和分子乘以

4

\frac{π}{6} = \frac{π 4}{6 \cdot 4} = \frac{π 4}{24}

对于

\frac{5 π}{8} :

将分母和分子乘以

3

\frac{5 π}{8} = \frac{5 π 3}{8 \cdot 3} = \frac{15 π}{24}

= - \frac{π 4}{24} + \frac{15 π}{24}

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 4 + 15 π}{24}

同类项相加:

- 4 π + 15 π = 11 π

= πn + \frac{11 π}{24}

θ = πn + \frac{11 π}{24}

θ = πn + \frac{11 π}{24}

θ = πn + \frac{11 π}{24}

解

2 (θ + \frac{π}{6}) = \frac{7 π}{4} + 2 πn : θ = πn + \frac{17 π}{24}

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

两边除以

2

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

两边除以

2

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

化简

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

化简

\frac{2 ( θ + \frac{π}{6} )}{2} : θ + \frac{π}{6}

\frac{2 ( θ + \frac{π}{6} )}{2}

数字相除:

\frac{2}{2} = 1

= θ + \frac{π}{6}

化简

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π}{8} + πn

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{4}}{2} = \frac{7 π}{8}

\frac{\frac{7 π}{4}}{2}

使用分式法则:

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

= \frac{7 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{7 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{7 π}{8} + πn

θ + \frac{π}{6} = \frac{7 π}{8} + πn

θ + \frac{π}{6} = \frac{7 π}{8} + πn

θ + \frac{π}{6} = \frac{7 π}{8} + πn

将

\frac{π}{6}

到右边

θ + \frac{π}{6} = \frac{7 π}{8} + πn

两边减去

\frac{π}{6}

θ + \frac{π}{6} - \frac{π}{6} = \frac{7 π}{8} + πn - \frac{π}{6}

化简

θ + \frac{π}{6} - \frac{π}{6} = \frac{7 π}{8} + πn - \frac{π}{6}

化简

θ + \frac{π}{6} - \frac{π}{6} : θ

θ + \frac{π}{6} - \frac{π}{6}

同类项相加:

\frac{π}{6} - \frac{π}{6} = 0

= θ

化简

\frac{7 π}{8} + πn - \frac{π}{6} : πn + \frac{17 π}{24}

\frac{7 π}{8} + πn - \frac{π}{6}

对同类项分组

= πn - \frac{π}{6} + \frac{7 π}{8}

6, 8

的最小公倍数:

24

6, 8

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

8

质因数分解:

2 \cdot 2 \cdot 2

8

8

除以

28 = 4 \cdot 2

= 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

将每个因子乘以它在

6

或

8

中出现的最多次数

= 2 \cdot 2 \cdot 2 \cdot 3

数字相乘:

2 \cdot 2 \cdot 2 \cdot 3 = 24

= 24

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

24

对于

\frac{π}{6} :

将分母和分子乘以

4

\frac{π}{6} = \frac{π 4}{6 \cdot 4} = \frac{π 4}{24}

对于

\frac{7 π}{8} :

将分母和分子乘以

3

\frac{7 π}{8} = \frac{7 π 3}{8 \cdot 3} = \frac{21 π}{24}

= - \frac{π 4}{24} + \frac{21 π}{24}

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 4 + 21 π}{24}

同类项相加:

- 4 π + 21 π = 17 π

= πn + \frac{17 π}{24}

θ = πn + \frac{17 π}{24}

θ = πn + \frac{17 π}{24}

θ = πn + \frac{17 π}{24}

θ = πn + \frac{11 π}{24}, θ = πn + \frac{17 π}{24}

作图

流行的例子

tan (x) = 0.2092

sin(θ+1)=cos(θ)

sin (θ + 1) = cos (θ)

0 = sin (x + c)

7sin(2θ)-2sin(θ)=0

7 sin (2 θ) - 2 sin (θ) = 0

cot(2x+pi/3)-sqrt(3)=0,-pi<x<pi

cot (2 x + \frac{π}{3}) - 3 = 0, - π < x < π