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受欢迎的三角函数 >

2sin^2(x)-3cos(2x)=3

解答

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

解答

x = \frac{π}{3} + πn, x = \frac{2 π}{3} + πn

+1

度数

x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

3

2 sin^{2} (x) - 3 cos (2 x) - 3 = 0

使用三角恒等式改写

- 3 + 2 sin^{2} (x) - 3 cos (2 x)

使用倍角公式:

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

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

= - 3 - (cos (2 x) - 1) - 3 cos (2 x)

化简

- 3 - (cos (2 x) - 1) - 3 cos (2 x) : - 4 cos (2 x) - 2

- 3 - (cos (2 x) - 1) - 3 cos (2 x)

- (cos (2 x) - 1) : - cos (2 x) + 1

- (cos (2 x) - 1)

打开括号

= - (cos (2 x)) - (- 1)

使用加减运算法则

- (- a) = a, - (a) = - a

= - cos (2 x) + 1

= - 3 - cos (2 x) + 1 - 3 cos (2 x)

化简

- 3 - cos (2 x) + 1 - 3 cos (2 x) : - 4 cos (2 x) - 2

- 3 - cos (2 x) + 1 - 3 cos (2 x)

对同类项分组

= - cos (2 x) - 3 cos (2 x) - 3 + 1

同类项相加:

- cos (2 x) - 3 cos (2 x) = - 4 cos (2 x)

= - 4 cos (2 x) - 3 + 1

数字相加/相减:

- 3 + 1 = - 2

= - 4 cos (2 x) - 2

= - 4 cos (2 x) - 2

= - 4 cos (2 x) - 2

- 2 - 4 cos (2 x) = 0

将

2

到右边

- 2 - 4 cos (2 x) = 0

两边加上

2

- 2 - 4 cos (2 x) + 2 = 0 + 2

化简

- 4 cos (2 x) = 2

- 4 cos (2 x) = 2

两边除以

- 4

- 4 cos (2 x) = 2

两边除以

- 4

\frac{- 4 cos ( 2 x )}{- 4} = \frac{2}{- 4}

化简

\frac{- 4 cos ( 2 x )}{- 4} = \frac{2}{- 4}

化简

\frac{- 4 cos ( 2 x )}{- 4} : cos (2 x)

\frac{- 4 cos ( 2 x )}{- 4}

使用分式法则:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4 cos ( 2 x )}{4}

数字相除:

\frac{4}{4} = 1

= cos (2 x)

化简

\frac{2}{- 4} : - \frac{1}{2}

\frac{2}{- 4}

使用分式法则:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

cos (2 x) = - \frac{1}{2}

cos (2 x) = - \frac{1}{2}

cos (2 x) = - \frac{1}{2}

cos (2 x) = - \frac{1}{2}

的通解

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

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

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

解

2 x = \frac{2 π}{3} + 2 πn : x = \frac{π}{3} + πn

2 x = \frac{2 π}{3} + 2 πn

两边除以

2

2 x = \frac{2 π}{3} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{3} + πn

\frac{\frac{2 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

使用分式法则:

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

= \frac{2 π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{2 π}{6}

约分:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

解

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

2 x = \frac{4 π}{3} + 2 πn

两边除以

2

2 x = \frac{4 π}{3} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

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

\frac{\frac{4 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{4 π}{3}}{2} = \frac{2 π}{3}

\frac{\frac{4 π}{3}}{2}

使用分式法则:

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

= \frac{4 π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{4 π}{6}

约分:

2

= \frac{2 π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{π}{3} + πn, x = \frac{2 π}{3} + πn

作图

流行的例子

cos(pi+x)+sin(x-pi/2)=1

cos (π + x) + sin (x - \frac{π}{2}) = 1

sin^2(a)+1/(sec(a))= 5/4

sin^{2} (a) + \frac{1}{sec ( a )} = \frac{5}{4}

(csc(x))/(cot(x))=sqrt(2)

\frac{csc ( x )}{cot ( x )} = 2

sin(4x)+6cos(2x)=0

sin (4 x) + 6 cos (2 x) = 0

sec(a)= 41/12 ,2pi<a<(5pi)/2 ,sin(a/2)

sec (a) = \frac{41}{12}, 2 π < a < \frac{5 π}{2}, sin (\frac{a}{2})