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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

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

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

使用倍角公式:

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

= - 1 + 2 cos^{2} (x) - 1 - 3 cos (x)

化简

- 1 + 2 cos^{2} (x) - 1 - 3 cos (x) : 2 cos^{2} (x) - 3 cos (x) - 2

- 1 + 2 cos^{2} (x) - 1 - 3 cos (x)

对同类项分组

= 2 cos^{2} (x) - 3 cos (x) - 1 - 1

数字相减:

- 1 - 1 = - 2

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

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

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

用替代法求解

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

令

: cos (x) = u

- 2 + 2 u^{2} - 3 u = 0

- 2 + 2 u^{2} - 3 u = 0 : u = 2, u = - \frac{1}{2}

- 2 + 2 u^{2} - 3 u = 0

改写成标准形式

a x^{2} + b x + c = 0

2 u^{2} - 3 u - 2 = 0

使用求根公式求解

2 u^{2} - 3 u - 2 = 0

二次方程求根公式:

若

a = 2, b = - 3, c = - 2

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

(- 3)^{2} - 4 \cdot 2 (- 2) = 5

(- 3)^{2} - 4 \cdot 2 (- 2)

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 2 \cdot 2

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 3)^{2} = 3^{2}

= 3^{2} + 4 \cdot 2 \cdot 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 5}{2 \cdot 2}, u_{2} = \frac{- ( - 3 ) - 5}{2 \cdot 2}

u = \frac{- ( - 3 ) + 5}{2 \cdot 2} : 2

\frac{- ( - 3 ) + 5}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 + 5}{2 \cdot 2}

数字相加:

3 + 5 = 8

= \frac{8}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= 2

u = \frac{- ( - 3 ) - 5}{2 \cdot 2} : - \frac{1}{2}

\frac{- ( - 3 ) - 5}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 - 5}{2 \cdot 2}

数字相减:

3 - 5 = - 2

= \frac{- 2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

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

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

cos (x) = 2 :

无解

cos (x) = 2

- 1 \leq cos (x) \leq 1

无解

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

cos (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}

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

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

合并所有解

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

作图

流行的例子

csc (θ) = 4

2sin(2θ)=-sqrt(2)

2 sin (2 θ) = - 2

2sin(2x)=sqrt(2)

2 sin (2 x) = 2

2 sec (θ) + 1 = 0

3arccos(2x)=2pi

3 arccos (2 x) = 2 π