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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

+1

度数

x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

求解步骤

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

两边减去

2

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

使用三角恒等式改写

- 2 + 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

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

化简

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

- 2 - (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

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

化简

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

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

对同类项分组

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

同类项相加:

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

= 2 cos (2 x) - 2 + 1

数字相加/相减:

- 2 + 1 = - 1

= 2 cos (2 x) - 1

= 2 cos (2 x) - 1

= 2 cos (2 x) - 1

- 1 + 2 cos (2 x) = 0

将

1

到右边

- 1 + 2 cos (2 x) = 0

两边加上

1

- 1 + 2 cos (2 x) + 1 = 0 + 1

化简

2 cos (2 x) = 1

2 cos (2 x) = 1

两边除以

2

2 cos (2 x) = 1

两边除以

2

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

化简

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

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

解

2 x = \frac{π}{3} + 2 πn : x = \frac{π}{6} + πn

2 x = \frac{π}{3} + 2 πn

两边除以

2

2 x = \frac{π}{3} + 2 πn

两边除以

2

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

化简

\frac{2 x}{2} = \frac{\frac{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{π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

使用分式法则:

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

= \frac{π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

解

2 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{6} + πn

2 x = \frac{5 π}{3} + 2 πn

两边除以

2

2 x = \frac{5 π}{3} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{5 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{5 π}{3}}{2} + \frac{2 πn}{2} : \frac{5 π}{6} + πn

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{3}}{2} = \frac{5 π}{6}

\frac{\frac{5 π}{3}}{2}

使用分式法则:

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

= \frac{5 π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{5 π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{π}{6} + πn, x = \frac{5 π}{6} + πn

作图

流行的例子

solvefor x,cos(3x)=1

so l v e f or x, cos (3 x) = 1

sec(x)+tan(x)=2

sec (x) + tan (x) = 2

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

sin (x) = cos (4 x)

sin^2(x)+sin(x)=0,0<= x<= 2pi

sin^{2} (x) + sin (x) = 0, 0 \leq x \leq 2 π