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受欢迎的三角函数 >

5cos^2(x)=6sin(x)

解答

5 cos^{2} (x) = 6 sin (x)

解答

x = 0.60187 \dots + 2 πn, x = π - 0.60187 \dots + 2 πn

+1

度数

x = 34.48499 \dots^{\circ} + 36 0^{\circ} n, x = 145.51500 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 cos^{2} (x) = 6 sin (x)

两边减去

6 sin (x)

5 cos^{2} (x) - 6 sin (x) = 0

使用三角恒等式改写

5 cos^{2} (x) - 6 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)

= 5 (1 - sin^{2} (x)) - 6 sin (x)

(1 - sin^{2} (x)) \cdot 5 - 6 sin (x) = 0

用替代法求解

(1 - sin^{2} (x)) \cdot 5 - 6 sin (x) = 0

令

: sin (x) = u

(1 - u^{2}) \cdot 5 - 6 u = 0

(1 - u^{2}) \cdot 5 - 6 u = 0 : u = - \frac{3 + 34}{5}, u = \frac{34 - 3}{5}

(1 - u^{2}) \cdot 5 - 6 u = 0

展开

(1 - u^{2}) \cdot 5 - 6 u : 5 - 5 u^{2} - 6 u

(1 - u^{2}) \cdot 5 - 6 u

= 5 (1 - u^{2}) - 6 u

乘开

5 (1 - u^{2}) : 5 - 5 u^{2}

5 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 5, b = 1, c = u^{2}

= 5 \cdot 1 - 5 u^{2}

数字相乘:

5 \cdot 1 = 5

= 5 - 5 u^{2}

= 5 - 5 u^{2} - 6 u

5 - 5 u^{2} - 6 u = 0

改写成标准形式

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

- 5 u^{2} - 6 u + 5 = 0

使用求根公式求解

- 5 u^{2} - 6 u + 5 = 0

二次方程求根公式:

若

a = - 5, b = - 6, c = 5

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 ( - 5 ) \cdot 5}{2 ( - 5 )}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 ( - 5 ) \cdot 5}{2 ( - 5 )}

(- 6)^{2} - 4 (- 5) \cdot 5 = 234

(- 6)^{2} - 4 (- 5) \cdot 5

使用法则

- (- a) = a

= (- 6)^{2} + 4 \cdot 5 \cdot 5

使用指数法则:

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

若

n

是偶数

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 5 \cdot 5

数字相乘:

4 \cdot 5 \cdot 5 = 100

= 6^{2} + 100

6^{2} = 36

= 36 + 100

数字相加:

36 + 100 = 136

= 136

136

质因数分解:

2^{3} \cdot 17

136

136

除以

2136 = 68 \cdot 2

= 2 \cdot 68

68

除以

268 = 34 \cdot 2

= 2 \cdot 2 \cdot 34

34

除以

234 = 17 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 17

2, 17

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 17

= 2^{3} \cdot 17

= 2^{3} \cdot 17

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2 \cdot 17

使用根式运算法则:

= 2^{2} 2 \cdot 17

使用根式运算法则:

2^{2} = 2

= 2 2 \cdot 17

整理后得

= 234

u_{1, 2} = \frac{- ( - 6 ) \pm 2 34}{2 ( - 5 )}

将解分隔开

u_{1} = \frac{- ( - 6 ) + 2 34}{2 ( - 5 )}, u_{2} = \frac{- ( - 6 ) - 2 34}{2 ( - 5 )}

u = \frac{- ( - 6 ) + 2 34}{2 ( - 5 )} : - \frac{3 + 34}{5}

\frac{- ( - 6 ) + 2 34}{2 ( - 5 )}

去除括号:

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

= \frac{6 + 2 34}{- 2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{6 + 2 34}{- 10}

使用分式法则:

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

= - \frac{6 + 2 34}{10}

消掉

\frac{6 + 2 34}{10} : \frac{3 + 34}{5}

\frac{6 + 2 34}{10}

分解

6 + 234 : 2 (3 + 34)

6 + 234

改写为

= 2 \cdot 3 + 234

因式分解出通项

2

= 2 (3 + 34)

= \frac{2 ( 3 + 34 )}{10}

约分:

2

= \frac{3 + 34}{5}

= - \frac{3 + 34}{5}

u = \frac{- ( - 6 ) - 2 34}{2 ( - 5 )} : \frac{34 - 3}{5}

\frac{- ( - 6 ) - 2 34}{2 ( - 5 )}

去除括号:

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

= \frac{6 - 2 34}{- 2 \cdot 5}

数字相乘:

2 \cdot 5 = 10

= \frac{6 - 2 34}{- 10}

使用分式法则:

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

6 - 234 = - (234 - 6)

= \frac{2 34 - 6}{10}

分解

234 - 6 : 2 (34 - 3)

234 - 6

改写为

= 234 - 2 \cdot 3

因式分解出通项

2

= 2 (34 - 3)

= \frac{2 ( 34 - 3 )}{10}

约分:

2

= \frac{34 - 3}{5}

二次方程组的解是:

u = - \frac{3 + 34}{5}, u = \frac{34 - 3}{5}

u = sin (x)

代回

sin (x) = - \frac{3 + 34}{5}, sin (x) = \frac{34 - 3}{5}

sin (x) = - \frac{3 + 34}{5}, sin (x) = \frac{34 - 3}{5}

sin (x) = - \frac{3 + 34}{5} :

无解

sin (x) = - \frac{3 + 34}{5}

- 1 \leq sin (x) \leq 1

无解

sin (x) = \frac{34 - 3}{5} : x = arcsin (\frac{34 - 3}{5}) + 2 πn, x = π - arcsin (\frac{34 - 3}{5}) + 2 πn

sin (x) = \frac{34 - 3}{5}

使用反三角函数性质

sin (x) = \frac{34 - 3}{5}

sin (x) = \frac{34 - 3}{5}

的通解

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

x = arcsin (\frac{34 - 3}{5}) + 2 πn, x = π - arcsin (\frac{34 - 3}{5}) + 2 πn

x = arcsin (\frac{34 - 3}{5}) + 2 πn, x = π - arcsin (\frac{34 - 3}{5}) + 2 πn

合并所有解

x = arcsin (\frac{34 - 3}{5}) + 2 πn, x = π - arcsin (\frac{34 - 3}{5}) + 2 πn

以小数形式表示解

x = 0.60187 \dots + 2 πn, x = π - 0.60187 \dots + 2 πn

作图

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