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受欢迎的三角函数 >

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

解答

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

解答

x = 0.58066 \dots + 2 πn, x = π - 0.58066 \dots + 2 πn

+1

度数

x = 33.26990 \dots^{\circ} + 36 0^{\circ} n, x = 146.73009 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

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

数字相乘:

2 \cdot 1 = 2

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

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

化简

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

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

对同类项分组

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

同类项相加:

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

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

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

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

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

用替代法求解

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

令

: sin (x) = u

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

2 - 2 u - 3 u^{2} = 0 : u = - \frac{1 + 7}{3}, u = \frac{7 - 1}{3}

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

(- 2)^{2} - 4 (- 3) \cdot 2 = 27

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 2^{2} + 24

2^{2} = 4

= 4 + 24

数字相加:

4 + 24 = 28

= 28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

u_{1, 2} = \frac{- ( - 2 ) \pm 2 7}{2 ( - 3 )}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 2 7}{2 ( - 3 )}, u_{2} = \frac{- ( - 2 ) - 2 7}{2 ( - 3 )}

u = \frac{- ( - 2 ) + 2 7}{2 ( - 3 )} : - \frac{1 + 7}{3}

\frac{- ( - 2 ) + 2 7}{2 ( - 3 )}

去除括号:

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

= \frac{2 + 2 7}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{2 + 2 7}{- 6}

使用分式法则:

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

= - \frac{2 + 2 7}{6}

消掉

\frac{2 + 2 7}{6} : \frac{1 + 7}{3}

\frac{2 + 2 7}{6}

分解

2 + 27 : 2 (1 + 7)

2 + 27

改写为

= 2 \cdot 1 + 27

因式分解出通项

2

= 2 (1 + 7)

= \frac{2 ( 1 + 7 )}{6}

约分:

2

= \frac{1 + 7}{3}

= - \frac{1 + 7}{3}

u = \frac{- ( - 2 ) - 2 7}{2 ( - 3 )} : \frac{7 - 1}{3}

\frac{- ( - 2 ) - 2 7}{2 ( - 3 )}

去除括号:

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

= \frac{2 - 2 7}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{2 - 2 7}{- 6}

使用分式法则:

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

2 - 27 = - (27 - 2)

= \frac{2 7 - 2}{6}

分解

27 - 2 : 2 (7 - 1)

27 - 2

改写为

= 27 - 2 \cdot 1

因式分解出通项

2

= 2 (7 - 1)

= \frac{2 ( 7 - 1 )}{6}

约分:

2

= \frac{7 - 1}{3}

二次方程组的解是:

u = - \frac{1 + 7}{3}, u = \frac{7 - 1}{3}

u = sin (x)

代回

sin (x) = - \frac{1 + 7}{3}, sin (x) = \frac{7 - 1}{3}

sin (x) = - \frac{1 + 7}{3}, sin (x) = \frac{7 - 1}{3}

sin (x) = - \frac{1 + 7}{3} :

无解

sin (x) = - \frac{1 + 7}{3}

- 1 \leq sin (x) \leq 1

无解

sin (x) = \frac{7 - 1}{3} : x = arcsin (\frac{7 - 1}{3}) + 2 πn, x = π - arcsin (\frac{7 - 1}{3}) + 2 πn

sin (x) = \frac{7 - 1}{3}

使用反三角函数性质

sin (x) = \frac{7 - 1}{3}

sin (x) = \frac{7 - 1}{3}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{7 - 1}{3}) + 2 πn, x = π - arcsin (\frac{7 - 1}{3}) + 2 πn

x = arcsin (\frac{7 - 1}{3}) + 2 πn, x = π - arcsin (\frac{7 - 1}{3}) + 2 πn

合并所有解

x = arcsin (\frac{7 - 1}{3}) + 2 πn, x = π - arcsin (\frac{7 - 1}{3}) + 2 πn

以小数形式表示解

x = 0.58066 \dots + 2 πn, x = π - 0.58066 \dots + 2 πn

作图

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