zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

-10cos(2x)-32cos(x)-22=0

解答

- 10 cos (2 x) - 32 cos (x) - 22 = 0

解答

x = π + 2 πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

+1

度数

x = 18 0^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = - 126.86989 \dots^{\circ} + 36 0^{\circ} n

求解步骤

- 10 cos (2 x) - 32 cos (x) - 22 = 0

使用三角恒等式改写

- 22 - 10 cos (2 x) - 32 cos (x)

使用倍角公式:

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

= - 22 - 10 (2 cos^{2} (x) - 1) - 32 cos (x)

化简

- 22 - 10 (2 cos^{2} (x) - 1) - 32 cos (x) : - 20 cos^{2} (x) - 32 cos (x) - 12

- 22 - 10 (2 cos^{2} (x) - 1) - 32 cos (x)

乘开

- 10 (2 cos^{2} (x) - 1) : - 20 cos^{2} (x) + 10

- 10 (2 cos^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = - 10, b = 2 cos^{2} (x), c = 1

= - 10 \cdot 2 cos^{2} (x) - (- 10) \cdot 1

使用加减运算法则

- (- a) = a

= - 10 \cdot 2 cos^{2} (x) + 10 \cdot 1

化简

- 10 \cdot 2 cos^{2} (x) + 10 \cdot 1 : - 20 cos^{2} (x) + 10

- 10 \cdot 2 cos^{2} (x) + 10 \cdot 1

数字相乘:

10 \cdot 2 = 20

= - 20 cos^{2} (x) + 10 \cdot 1

数字相乘:

10 \cdot 1 = 10

= - 20 cos^{2} (x) + 10

= - 20 cos^{2} (x) + 10

= - 22 - 20 cos^{2} (x) + 10 - 32 cos (x)

化简

- 22 - 20 cos^{2} (x) + 10 - 32 cos (x) : - 20 cos^{2} (x) - 32 cos (x) - 12

- 22 - 20 cos^{2} (x) + 10 - 32 cos (x)

对同类项分组

= - 20 cos^{2} (x) - 32 cos (x) - 22 + 10

数字相加/相减:

- 22 + 10 = - 12

= - 20 cos^{2} (x) - 32 cos (x) - 12

= - 20 cos^{2} (x) - 32 cos (x) - 12

= - 20 cos^{2} (x) - 32 cos (x) - 12

- 12 - 20 cos^{2} (x) - 32 cos (x) = 0

用替代法求解

- 12 - 20 cos^{2} (x) - 32 cos (x) = 0

令

: cos (x) = u

- 12 - 20 u^{2} - 32 u = 0

- 12 - 20 u^{2} - 32 u = 0 : u = - 1, u = - \frac{3}{5}

- 12 - 20 u^{2} - 32 u = 0

改写成标准形式

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

- 20 u^{2} - 32 u - 12 = 0

使用求根公式求解

- 20 u^{2} - 32 u - 12 = 0

二次方程求根公式:

若

a = - 20, b = - 32, c = - 12

u_{1, 2} = \frac{- ( - 32 ) \pm ( - 32 ) ^{2} - 4 ( - 20 ) ( - 12 )}{2 ( - 20 )}

u_{1, 2} = \frac{- ( - 32 ) \pm ( - 32 ) ^{2} - 4 ( - 20 ) ( - 12 )}{2 ( - 20 )}

(- 32)^{2} - 4 (- 20) (- 12) = 8

(- 32)^{2} - 4 (- 20) (- 12)

使用法则

- (- a) = a

= (- 32)^{2} - 4 \cdot 20 \cdot 12

使用指数法则:

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

若

n

是偶数

(- 32)^{2} = 3 2^{2}

= 3 2^{2} - 4 \cdot 20 \cdot 12

数字相乘:

4 \cdot 20 \cdot 12 = 960

= 3 2^{2} - 960

3 2^{2} = 1024

= 1024 - 960

数字相减:

1024 - 960 = 64

= 64

因式分解数字:

64 = 8^{2}

= 8^{2}

使用根式运算法则:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 32 ) \pm 8}{2 ( - 20 )}

将解分隔开

u_{1} = \frac{- ( - 32 ) + 8}{2 ( - 20 )}, u_{2} = \frac{- ( - 32 ) - 8}{2 ( - 20 )}

u = \frac{- ( - 32 ) + 8}{2 ( - 20 )} : - 1

\frac{- ( - 32 ) + 8}{2 ( - 20 )}

去除括号:

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

= \frac{32 + 8}{- 2 \cdot 20}

数字相加:

32 + 8 = 40

= \frac{40}{- 2 \cdot 20}

数字相乘:

2 \cdot 20 = 40

= \frac{40}{- 40}

使用分式法则:

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

= - \frac{40}{40}

使用法则

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 32 ) - 8}{2 ( - 20 )} : - \frac{3}{5}

\frac{- ( - 32 ) - 8}{2 ( - 20 )}

去除括号:

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

= \frac{32 - 8}{- 2 \cdot 20}

数字相减:

32 - 8 = 24

= \frac{24}{- 2 \cdot 20}

数字相乘:

2 \cdot 20 = 40

= \frac{24}{- 40}

使用分式法则:

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

= - \frac{24}{40}

约分:

8

= - \frac{3}{5}

二次方程组的解是:

u = - 1, u = - \frac{3}{5}

u = cos (x)

代回

cos (x) = - 1, cos (x) = - \frac{3}{5}

cos (x) = - 1, cos (x) = - \frac{3}{5}

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

cos (x) = - \frac{3}{5} : x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

cos (x) = - \frac{3}{5}

使用反三角函数性质

cos (x) = - \frac{3}{5}

cos (x) = - \frac{3}{5}

的通解

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

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

合并所有解

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

以小数形式表示解

x = π + 2 πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

作图

流行的例子

tan(x)*sec(x)=sqrt(2)

tan (x) \cdot sec (x) = 2

cos(2x+pi/3)=sin(2x)

cos (2 x + \frac{π}{3}) = sin (2 x)

cot (2 t) = 0

tan(θ)=cot(θ),0<= θ<2pi

tan (θ) = cot (θ), 0 \leq θ < 2 π

sec (θ) = 1.984