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受欢迎的三角函数 >

12cos^2(x)-6=sin(x)

解答

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

解答

x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

+1

度数

x = - 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 228.59037 \dots^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

sin (x)

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

使用三角恒等式改写

- 6 - sin (x) + 12 cos^{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)

= - 6 - sin (x) + 12 (1 - sin^{2} (x))

化简

- 6 - sin (x) + 12 (1 - sin^{2} (x)) : - 12 sin^{2} (x) - sin (x) + 6

- 6 - sin (x) + 12 (1 - sin^{2} (x))

乘开

12 (1 - sin^{2} (x)) : 12 - 12 sin^{2} (x)

12 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 12, b = 1, c = sin^{2} (x)

= 12 \cdot 1 - 12 sin^{2} (x)

数字相乘:

12 \cdot 1 = 12

= 12 - 12 sin^{2} (x)

= - 6 - sin (x) + 12 - 12 sin^{2} (x)

化简

- 6 - sin (x) + 12 - 12 sin^{2} (x) : - 12 sin^{2} (x) - sin (x) + 6

- 6 - sin (x) + 12 - 12 sin^{2} (x)

对同类项分组

= - sin (x) - 12 sin^{2} (x) - 6 + 12

数字相加/相减:

- 6 + 12 = 6

= - 12 sin^{2} (x) - sin (x) + 6

= - 12 sin^{2} (x) - sin (x) + 6

= - 12 sin^{2} (x) - sin (x) + 6

6 - sin (x) - 12 sin^{2} (x) = 0

用替代法求解

6 - sin (x) - 12 sin^{2} (x) = 0

令

: sin (x) = u

6 - u - 12 u^{2} = 0

6 - u - 12 u^{2} = 0 : u = - \frac{3}{4}, u = \frac{2}{3}

6 - u - 12 u^{2} = 0

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 12, b = - 1, c = 6

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 12 ) \cdot 6}{2 ( - 12 )}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 ( - 12 ) \cdot 6}{2 ( - 12 )}

(- 1)^{2} - 4 (- 12) \cdot 6 = 17

(- 1)^{2} - 4 (- 12) \cdot 6

使用法则

- (- a) = a

= (- 1)^{2} + 4 \cdot 12 \cdot 6

(- 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 12 \cdot 6 = 288

4 \cdot 12 \cdot 6

数字相乘:

4 \cdot 12 \cdot 6 = 288

= 288

= 1 + 288

数字相加:

1 + 288 = 289

= 289

因式分解数字:

289 = 1 7^{2}

= 1 7^{2}

使用根式运算法则:

n a^{n} = a

1 7^{2} = 17

= 17

u_{1, 2} = \frac{- ( - 1 ) \pm 17}{2 ( - 12 )}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 17}{2 ( - 12 )}, u_{2} = \frac{- ( - 1 ) - 17}{2 ( - 12 )}

u = \frac{- ( - 1 ) + 17}{2 ( - 12 )} : - \frac{3}{4}

\frac{- ( - 1 ) + 17}{2 ( - 12 )}

去除括号:

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

= \frac{1 + 17}{- 2 \cdot 12}

数字相加:

1 + 17 = 18

= \frac{18}{- 2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{18}{- 24}

使用分式法则:

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

= - \frac{18}{24}

约分:

6

= - \frac{3}{4}

u = \frac{- ( - 1 ) - 17}{2 ( - 12 )} : \frac{2}{3}

\frac{- ( - 1 ) - 17}{2 ( - 12 )}

去除括号:

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

= \frac{1 - 17}{- 2 \cdot 12}

数字相减:

1 - 17 = - 16

= \frac{- 16}{- 2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{- 16}{- 24}

使用分式法则:

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

= \frac{16}{24}

约分:

8

= \frac{2}{3}

二次方程组的解是:

u = - \frac{3}{4}, u = \frac{2}{3}

u = sin (x)

代回

sin (x) = - \frac{3}{4}, sin (x) = \frac{2}{3}

sin (x) = - \frac{3}{4}, sin (x) = \frac{2}{3}

sin (x) = - \frac{3}{4} : x = arcsin (- \frac{3}{4}) + 2 πn, x = π + arcsin (\frac{3}{4}) + 2 πn

sin (x) = - \frac{3}{4}

使用反三角函数性质

sin (x) = - \frac{3}{4}

sin (x) = - \frac{3}{4}

的通解

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

x = arcsin (- \frac{3}{4}) + 2 πn, x = π + arcsin (\frac{3}{4}) + 2 πn

x = arcsin (- \frac{3}{4}) + 2 πn, x = π + arcsin (\frac{3}{4}) + 2 πn

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

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

使用反三角函数性质

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

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

的通解

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

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

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

合并所有解

x = arcsin (- \frac{3}{4}) + 2 πn, x = π + arcsin (\frac{3}{4}) + 2 πn, x = arcsin (\frac{2}{3}) + 2 πn, x = π - arcsin (\frac{2}{3}) + 2 πn

以小数形式表示解

x = - 0.84806 \dots + 2 πn, x = π + 0.84806 \dots + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

作图

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