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受欢迎的三角函数 >

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

解答

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

解答

x = 0.56024 \dots + 2 πn, x = 2 π - 0.56024 \dots + 2 πn

+1

度数

x = 32.09944 \dots^{\circ} + 36 0^{\circ} n, x = 327.90055 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

cos (x)

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

使用三角恒等式改写

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

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

- cos (x) + (1 - cos^{2} (x)) \cdot 3 = 0

用替代法求解

- cos (x) + (1 - cos^{2} (x)) \cdot 3 = 0

令

: cos (x) = u

- u + (1 - u^{2}) \cdot 3 = 0

- u + (1 - u^{2}) \cdot 3 = 0 : u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

- u + (1 - u^{2}) \cdot 3 = 0

展开

- u + (1 - u^{2}) \cdot 3 : - u + 3 - 3 u^{2}

- u + (1 - u^{2}) \cdot 3

= - u + 3 (1 - u^{2})

乘开

3 (1 - u^{2}) : 3 - 3 u^{2}

3 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = u^{2}

= 3 \cdot 1 - 3 u^{2}

数字相乘:

3 \cdot 1 = 3

= 3 - 3 u^{2}

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

(- 1)^{2} - 4 (- 3) \cdot 3 = 37

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

使用法则

- (- a) = a

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

(- 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 3 \cdot 3 = 36

4 \cdot 3 \cdot 3

数字相乘:

4 \cdot 3 \cdot 3 = 36

= 36

= 1 + 36

数字相加:

1 + 36 = 37

= 37

u_{1, 2} = \frac{- ( - 1 ) \pm 37}{2 ( - 3 )}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 37}{2 ( - 3 )}, u_{2} = \frac{- ( - 1 ) - 37}{2 ( - 3 )}

u = \frac{- ( - 1 ) + 37}{2 ( - 3 )} : - \frac{1 + 37}{6}

\frac{- ( - 1 ) + 37}{2 ( - 3 )}

去除括号:

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

= \frac{1 + 37}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{1 + 37}{- 6}

使用分式法则:

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

= - \frac{1 + 37}{6}

u = \frac{- ( - 1 ) - 37}{2 ( - 3 )} : \frac{37 - 1}{6}

\frac{- ( - 1 ) - 37}{2 ( - 3 )}

去除括号:

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

= \frac{1 - 37}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{1 - 37}{- 6}

使用分式法则:

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

1 - 37 = - (37 - 1)

= \frac{37 - 1}{6}

二次方程组的解是:

u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

u = cos (x)

代回

cos (x) = - \frac{1 + 37}{6}, cos (x) = \frac{37 - 1}{6}

cos (x) = - \frac{1 + 37}{6}, cos (x) = \frac{37 - 1}{6}

cos (x) = - \frac{1 + 37}{6} :

无解

cos (x) = - \frac{1 + 37}{6}

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{37 - 1}{6} : x = arccos (\frac{37 - 1}{6}) + 2 πn, x = 2 π - arccos (\frac{37 - 1}{6}) + 2 πn

cos (x) = \frac{37 - 1}{6}

使用反三角函数性质

cos (x) = \frac{37 - 1}{6}

cos (x) = \frac{37 - 1}{6}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{37 - 1}{6}) + 2 πn, x = 2 π - arccos (\frac{37 - 1}{6}) + 2 πn

x = arccos (\frac{37 - 1}{6}) + 2 πn, x = 2 π - arccos (\frac{37 - 1}{6}) + 2 πn

合并所有解

x = arccos (\frac{37 - 1}{6}) + 2 πn, x = 2 π - arccos (\frac{37 - 1}{6}) + 2 πn

以小数形式表示解

x = 0.56024 \dots + 2 πn, x = 2 π - 0.56024 \dots + 2 πn

作图

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