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受欢迎的三角函数 >

8cos^2(x)-12sin(x)-12=0

解答

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

解答

x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 27 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 12 - 12 sin (x) + 8 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)

= - 12 - 12 sin (x) + 8 (1 - sin^{2} (x))

化简

- 12 - 12 sin (x) + 8 (1 - sin^{2} (x)) : - 8 sin^{2} (x) - 12 sin (x) - 4

- 12 - 12 sin (x) + 8 (1 - sin^{2} (x))

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = 8, b = 1, c = sin^{2} (x)

= 8 \cdot 1 - 8 sin^{2} (x)

数字相乘:

8 \cdot 1 = 8

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

= - 12 - 12 sin (x) + 8 - 8 sin^{2} (x)

化简

- 12 - 12 sin (x) + 8 - 8 sin^{2} (x) : - 8 sin^{2} (x) - 12 sin (x) - 4

- 12 - 12 sin (x) + 8 - 8 sin^{2} (x)

对同类项分组

= - 12 sin (x) - 8 sin^{2} (x) - 12 + 8

数字相加/相减:

- 12 + 8 = - 4

= - 8 sin^{2} (x) - 12 sin (x) - 4

= - 8 sin^{2} (x) - 12 sin (x) - 4

= - 8 sin^{2} (x) - 12 sin (x) - 4

- 4 - 12 sin (x) - 8 sin^{2} (x) = 0

用替代法求解

- 4 - 12 sin (x) - 8 sin^{2} (x) = 0

令

: sin (x) = u

- 4 - 12 u - 8 u^{2} = 0

- 4 - 12 u - 8 u^{2} = 0 : u = - 1, u = - \frac{1}{2}

- 4 - 12 u - 8 u^{2} = 0

改写成标准形式

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

- 8 u^{2} - 12 u - 4 = 0

使用求根公式求解

- 8 u^{2} - 12 u - 4 = 0

二次方程求根公式:

若

a = - 8, b = - 12, c = - 4

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 ( - 8 ) ( - 4 )}{2 ( - 8 )}

u_{1, 2} = \frac{- ( - 12 ) \pm ( - 12 ) ^{2} - 4 ( - 8 ) ( - 4 )}{2 ( - 8 )}

(- 12)^{2} - 4 (- 8) (- 4) = 4

(- 12)^{2} - 4 (- 8) (- 4)

使用法则

- (- a) = a

= (- 12)^{2} - 4 \cdot 8 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 12)^{2} = 1 2^{2}

= 1 2^{2} - 4 \cdot 8 \cdot 4

数字相乘:

4 \cdot 8 \cdot 4 = 128

= 1 2^{2} - 128

1 2^{2} = 144

= 144 - 128

数字相减:

144 - 128 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- ( - 12 ) \pm 4}{2 ( - 8 )}

将解分隔开

u_{1} = \frac{- ( - 12 ) + 4}{2 ( - 8 )}, u_{2} = \frac{- ( - 12 ) - 4}{2 ( - 8 )}

u = \frac{- ( - 12 ) + 4}{2 ( - 8 )} : - 1

\frac{- ( - 12 ) + 4}{2 ( - 8 )}

去除括号:

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

= \frac{12 + 4}{- 2 \cdot 8}

数字相加:

12 + 4 = 16

= \frac{16}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{16}{- 16}

使用分式法则:

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

= - \frac{16}{16}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 12 ) - 4}{2 ( - 8 )} : - \frac{1}{2}

\frac{- ( - 12 ) - 4}{2 ( - 8 )}

去除括号:

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

= \frac{12 - 4}{- 2 \cdot 8}

数字相减:

12 - 4 = 8

= \frac{8}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{8}{- 16}

使用分式法则:

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

= - \frac{8}{16}

约分:

8

= - \frac{1}{2}

二次方程组的解是:

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

u = sin (x)

代回

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

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

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

sin (x) = - 1

sin (x) = - 1

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

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

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

合并所有解

x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

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