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受欢迎的三角函数 >

cos^2(x)=8sin^2(x)-6sin(x)

解答

cos^{2} (x) = 8 sin^{2} (x) - 6 sin (x)

解答

x = - 0.13851 \dots + 2 πn, x = π + 0.13851 \dots + 2 πn, x = 0.93523 \dots + 2 πn, x = π - 0.93523 \dots + 2 πn

+1

度数

x = - 7.93624 \dots^{\circ} + 36 0^{\circ} n, x = 187.93624 \dots^{\circ} + 36 0^{\circ} n, x = 53.58494 \dots^{\circ} + 36 0^{\circ} n, x = 126.41505 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) = 8 sin^{2} (x) - 6 sin (x)

两边减去

8 sin^{2} (x) - 6 sin (x)

cos^{2} (x) - 8 sin^{2} (x) + 6 sin (x) = 0

使用三角恒等式改写

cos^{2} (x) + 6 sin (x) - 8 sin^{2} (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)

= 1 - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x)

化简

1 - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x) : - 9 sin^{2} (x) + 6 sin (x) + 1

1 - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x)

对同类项分组

= - sin^{2} (x) + 6 sin (x) - 8 sin^{2} (x) + 1

同类项相加:

- sin^{2} (x) - 8 sin^{2} (x) = - 9 sin^{2} (x)

= - 9 sin^{2} (x) + 6 sin (x) + 1

= - 9 sin^{2} (x) + 6 sin (x) + 1

1 + 6 sin (x) - 9 sin^{2} (x) = 0

用替代法求解

1 + 6 sin (x) - 9 sin^{2} (x) = 0

令

: sin (x) = u

1 + 6 u - 9 u^{2} = 0

1 + 6 u - 9 u^{2} = 0 : u = - \frac{- 1 + 2}{3}, u = \frac{1 + 2}{3}

1 + 6 u - 9 u^{2} = 0

改写成标准形式

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

- 9 u^{2} + 6 u + 1 = 0

使用求根公式求解

- 9 u^{2} + 6 u + 1 = 0

二次方程求根公式:

若

a = - 9, b = 6, c = 1

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 9 ) \cdot 1}{2 ( - 9 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 9 ) \cdot 1}{2 ( - 9 )}

6^{2} - 4 (- 9) \cdot 1 = 62

6^{2} - 4 (- 9) \cdot 1

使用法则

- (- a) = a

= 6^{2} + 4 \cdot 9 \cdot 1

数字相乘:

4 \cdot 9 \cdot 1 = 36

= 6^{2} + 36

6^{2} = 36

= 36 + 36

数字相加:

36 + 36 = 72

= 72

72

质因数分解:

2^{3} \cdot 3^{2}

72

72

除以

272 = 36 \cdot 2

= 2 \cdot 36

36

除以

236 = 18 \cdot 2

= 2 \cdot 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

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

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

使用指数法则:

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

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

使用根式运算法则:

n ab = n a n b

= 2 2^{2} 3^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 2 \cdot 32

整理后得

= 62

u_{1, 2} = \frac{- 6 \pm 6 2}{2 ( - 9 )}

将解分隔开

u_{1} = \frac{- 6 + 6 2}{2 ( - 9 )}, u_{2} = \frac{- 6 - 6 2}{2 ( - 9 )}

u = \frac{- 6 + 6 2}{2 ( - 9 )} : - \frac{- 1 + 2}{3}

\frac{- 6 + 6 2}{2 ( - 9 )}

去除括号:

(- a) = - a

= \frac{- 6 + 6 2}{- 2 \cdot 9}

数字相乘:

2 \cdot 9 = 18

= \frac{- 6 + 6 2}{- 18}

使用分式法则:

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

= - \frac{- 6 + 6 2}{18}

消掉

\frac{- 6 + 6 2}{18} : \frac{2 - 1}{3}

\frac{- 6 + 6 2}{18}

分解

- 6 + 62 : 6 (- 1 + 2)

- 6 + 62

改写为

= - 6 \cdot 1 + 62

因式分解出通项

6

= 6 (- 1 + 2)

= \frac{6 ( - 1 + 2 )}{18}

约分:

6

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

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

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

u = \frac{- 6 - 6 2}{2 ( - 9 )} : \frac{1 + 2}{3}

\frac{- 6 - 6 2}{2 ( - 9 )}

去除括号:

(- a) = - a

= \frac{- 6 - 6 2}{- 2 \cdot 9}

数字相乘:

2 \cdot 9 = 18

= \frac{- 6 - 6 2}{- 18}

使用分式法则:

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

- 6 - 62 = - (6 + 62)

= \frac{6 + 6 2}{18}

分解

6 + 62 : 6 (1 + 2)

6 + 62

改写为

= 6 \cdot 1 + 62

因式分解出通项

6

= 6 (1 + 2)

= \frac{6 ( 1 + 2 )}{18}

约分:

6

= \frac{1 + 2}{3}

二次方程组的解是:

u = - \frac{- 1 + 2}{3}, u = \frac{1 + 2}{3}

u = sin (x)

代回

sin (x) = - \frac{- 1 + 2}{3}, sin (x) = \frac{1 + 2}{3}

sin (x) = - \frac{- 1 + 2}{3}, sin (x) = \frac{1 + 2}{3}

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

sin (x) = - \frac{- 1 + 2}{3}

使用反三角函数性质

sin (x) = - \frac{- 1 + 2}{3}

sin (x) = - \frac{- 1 + 2}{3}

的通解

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

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

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

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

sin (x) = \frac{1 + 2}{3}

使用反三角函数性质

sin (x) = \frac{1 + 2}{3}

sin (x) = \frac{1 + 2}{3}

的通解

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

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

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

合并所有解

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

以小数形式表示解

x = - 0.13851 \dots + 2 πn, x = π + 0.13851 \dots + 2 πn, x = 0.93523 \dots + 2 πn, x = π - 0.93523 \dots + 2 πn

作图

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