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受欢迎的三角函数 >

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

解答

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

解答

x = π + 2 πn

+1

度数

x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

- 2

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

用替代法求解

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

令

: cos (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 1, b = 2, c = 3

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

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

2^{2} - 4 (- 1) \cdot 3 = 4

2^{2} - 4 (- 1) \cdot 3

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 3

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

数字相加:

4 + 12 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

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

将解分隔开

u_{1} = \frac{- 2 + 4}{2 ( - 1 )}, u_{2} = \frac{- 2 - 4}{2 ( - 1 )}

u = \frac{- 2 + 4}{2 ( - 1 )} : - 1

\frac{- 2 + 4}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 2 + 4}{- 2 \cdot 1}

数字相加/相减:

- 2 + 4 = 2

= \frac{2}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{2}{- 2}

使用分式法则:

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

= - \frac{2}{2}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- 2 - 4}{2 ( - 1 )} : 3

\frac{- 2 - 4}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 2 - 4}{- 2 \cdot 1}

数字相减:

- 2 - 4 = - 6

= \frac{- 6}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6}{- 2}

使用分式法则:

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

= \frac{6}{2}

数字相除:

\frac{6}{2} = 3

= 3

二次方程组的解是:

u = - 1, u = 3

u = cos (x)

代回

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

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

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

cos (x) = 3 :

无解

cos (x) = 3

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = π + 2 πn

作图

流行的例子

sqrt(7)sin(θ)-1=0

7 sin (θ) - 1 = 0

sin(3θ)=-1/(sqrt(2))

sin (3 θ) = - \frac{1}{2}

2sin(x)-cot(x)-csc(x)=0

2 sin (x) - cot (x) - csc (x) = 0

(-tan^2(x))/2 =cos^2(x)-1

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

sin (a) = \frac{π}{4}