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受欢迎的三角函数 >

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

解答

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

解答

x \in R 无解

求解步骤

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

使用三角恒等式改写

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

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

使用倍角公式:

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

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

化简

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

- 1 - sin (x) - 2 sin^{2} (x) = 0

用替代法求解

- 1 - sin (x) - 2 sin^{2} (x) = 0

令

: sin (x) = u

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

- 1 - u - 2 u^{2} = 0 : u = - \frac{1}{4} - i \frac{7}{4}, u = - \frac{1}{4} + i \frac{7}{4}

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

改写成标准形式

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

- 2 u^{2} - u - 1 = 0

使用求根公式求解

- 2 u^{2} - u - 1 = 0

二次方程求根公式:

若

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

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

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

化简

(- 1)^{2} - 4 (- 2) (- 1) : 7 i

(- 1)^{2} - 4 (- 2) (- 1)

使用法则

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 8

= 1 - 8

数字相减:

1 - 8 = - 7

= - 7

使用根式运算法则:

- a = - 1 a

- 7 = - 1 7

= - 1 7

使用虚数运算法则:

- 1 = i

= 7 i

u_{1, 2} = \frac{- ( - 1 ) \pm 7 i}{2 ( - 2 )}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 7 i}{2 ( - 2 )}, u_{2} = \frac{- ( - 1 ) - 7 i}{2 ( - 2 )}

u = \frac{- ( - 1 ) + 7 i}{2 ( - 2 )} : - \frac{1}{4} - i \frac{7}{4}

\frac{- ( - 1 ) + 7 i}{2 ( - 2 )}

去除括号:

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

= \frac{1 + 7 i}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{1 + 7 i}{- 4}

使用分式法则:

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

= - \frac{1 + 7 i}{4}

将

- \frac{1 + 7 i}{4}

改写成标准复数形式:

- \frac{1}{4} - \frac{7}{4} i

- \frac{1 + 7 i}{4}

使用分式法则:

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

\frac{1 + 7 i}{4} = - (\frac{1}{4}) - (\frac{7 i}{4})

= - (\frac{1}{4}) - (\frac{7 i}{4})

去除括号:

(a) = a

= - \frac{1}{4} - \frac{7 i}{4}

= - \frac{1}{4} - \frac{7}{4} i

u = \frac{- ( - 1 ) - 7 i}{2 ( - 2 )} : - \frac{1}{4} + i \frac{7}{4}

\frac{- ( - 1 ) - 7 i}{2 ( - 2 )}

去除括号:

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

= \frac{1 - 7 i}{- 2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{1 - 7 i}{- 4}

使用分式法则:

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

= - \frac{1 - 7 i}{4}

将

- \frac{1 - 7 i}{4}

改写成标准复数形式:

- \frac{1}{4} + \frac{7}{4} i

- \frac{1 - 7 i}{4}

使用分式法则:

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

\frac{1 - 7 i}{4} = - (\frac{1}{4}) - (- \frac{7 i}{4})

= - (\frac{1}{4}) - (- \frac{7 i}{4})

去除括号:

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

= - \frac{1}{4} + \frac{7 i}{4}

= - \frac{1}{4} + \frac{7}{4} i

二次方程组的解是:

u = - \frac{1}{4} - i \frac{7}{4}, u = - \frac{1}{4} + i \frac{7}{4}

u = sin (x)

代回

sin (x) = - \frac{1}{4} - i \frac{7}{4}, sin (x) = - \frac{1}{4} + i \frac{7}{4}

sin (x) = - \frac{1}{4} - i \frac{7}{4}, sin (x) = - \frac{1}{4} + i \frac{7}{4}

sin (x) = - \frac{1}{4} - i \frac{7}{4} :

无解

sin (x) = - \frac{1}{4} - i \frac{7}{4}

无解

sin (x) = - \frac{1}{4} + i \frac{7}{4} :

无解

sin (x) = - \frac{1}{4} + i \frac{7}{4}

无解

合并所有解

x \in R 无解

作图

流行的例子

sec(θ)=tan(θ)

sec (θ) = tan (θ)

-sqrt(3)csc(θ)=2

- 3 csc (θ) = 2

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

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

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

- 2 + 2 cos (x) + 4 cos^{2} (x) = 0

sin(A)=(6.347)/(12.7)

sin (A) = \frac{6.347}{12.7}