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 >

8cos(2x)=2sin(x)+7

解答

8 cos (2 x) = 2 sin (x) + 7

解答

x = - 0.32593 \dots + 2 πn, x = π + 0.32593 \dots + 2 πn, x = 0.19645 \dots + 2 πn, x = π - 0.19645 \dots + 2 πn

+1

度数

x = - 18.67466 \dots^{\circ} + 36 0^{\circ} n, x = 198.67466 \dots^{\circ} + 36 0^{\circ} n, x = 11.25606 \dots^{\circ} + 36 0^{\circ} n, x = 168.74393 \dots^{\circ} + 36 0^{\circ} n

求解步骤

8 cos (2 x) = 2 sin (x) + 7

两边减去

2 sin (x) + 7

8 cos (2 x) - 2 sin (x) - 7 = 0

使用三角恒等式改写

- 7 - 2 sin (x) + 8 cos (2 x)

使用倍角公式:

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

= - 7 - 2 sin (x) + 8 (1 - 2 sin^{2} (x))

化简

- 7 - 2 sin (x) + 8 (1 - 2 sin^{2} (x)) : - 16 sin^{2} (x) - 2 sin (x) + 1

- 7 - 2 sin (x) + 8 (1 - 2 sin^{2} (x))

乘开

8 (1 - 2 sin^{2} (x)) : 8 - 16 sin^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

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

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

化简

8 \cdot 1 - 8 \cdot 2 sin^{2} (x) : 8 - 16 sin^{2} (x)

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

数字相乘:

8 \cdot 1 = 8

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

数字相乘:

8 \cdot 2 = 16

= 8 - 16 sin^{2} (x)

= 8 - 16 sin^{2} (x)

= - 7 - 2 sin (x) + 8 - 16 sin^{2} (x)

化简

- 7 - 2 sin (x) + 8 - 16 sin^{2} (x) : - 16 sin^{2} (x) - 2 sin (x) + 1

- 7 - 2 sin (x) + 8 - 16 sin^{2} (x)

对同类项分组

= - 2 sin (x) - 16 sin^{2} (x) - 7 + 8

数字相加/相减:

- 7 + 8 = 1

= - 16 sin^{2} (x) - 2 sin (x) + 1

= - 16 sin^{2} (x) - 2 sin (x) + 1

= - 16 sin^{2} (x) - 2 sin (x) + 1

1 - 16 sin^{2} (x) - 2 sin (x) = 0

用替代法求解

1 - 16 sin^{2} (x) - 2 sin (x) = 0

令

: sin (x) = u

1 - 16 u^{2} - 2 u = 0

1 - 16 u^{2} - 2 u = 0 : u = - \frac{1 + 17}{16}, u = \frac{17 - 1}{16}

1 - 16 u^{2} - 2 u = 0

改写成标准形式

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

- 16 u^{2} - 2 u + 1 = 0

使用求根公式求解

- 16 u^{2} - 2 u + 1 = 0

二次方程求根公式:

若

a = - 16, b = - 2, c = 1

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 16 ) \cdot 1}{2 ( - 16 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 16 ) \cdot 1}{2 ( - 16 )}

(- 2)^{2} - 4 (- 16) \cdot 1 = 217

(- 2)^{2} - 4 (- 16) \cdot 1

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 16 \cdot 1

使用指数法则:

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

若

n

是偶数

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

= 2^{2} + 4 \cdot 16 \cdot 1

数字相乘:

4 \cdot 16 \cdot 1 = 64

= 2^{2} + 64

2^{2} = 4

= 4 + 64

数字相加:

4 + 64 = 68

= 68

68

质因数分解:

2^{2} \cdot 17

68

68

除以

268 = 34 \cdot 2

= 2 \cdot 34

34

除以

234 = 17 \cdot 2

= 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 17

= 2^{2} \cdot 17

= 2^{2} \cdot 17

使用根式运算法则:

= 17 2^{2}

使用根式运算法则:

2^{2} = 2

= 217

u_{1, 2} = \frac{- ( - 2 ) \pm 2 17}{2 ( - 16 )}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 2 17}{2 ( - 16 )}, u_{2} = \frac{- ( - 2 ) - 2 17}{2 ( - 16 )}

u = \frac{- ( - 2 ) + 2 17}{2 ( - 16 )} : - \frac{1 + 17}{16}

\frac{- ( - 2 ) + 2 17}{2 ( - 16 )}

去除括号:

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

= \frac{2 + 2 17}{- 2 \cdot 16}

数字相乘:

2 \cdot 16 = 32

= \frac{2 + 2 17}{- 32}

使用分式法则:

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

= - \frac{2 + 2 17}{32}

消掉

\frac{2 + 2 17}{32} : \frac{1 + 17}{16}

\frac{2 + 2 17}{32}

分解

2 + 217 : 2 (1 + 17)

2 + 217

改写为

= 2 \cdot 1 + 217

因式分解出通项

2

= 2 (1 + 17)

= \frac{2 ( 1 + 17 )}{32}

约分:

2

= \frac{1 + 17}{16}

= - \frac{1 + 17}{16}

u = \frac{- ( - 2 ) - 2 17}{2 ( - 16 )} : \frac{17 - 1}{16}

\frac{- ( - 2 ) - 2 17}{2 ( - 16 )}

去除括号:

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

= \frac{2 - 2 17}{- 2 \cdot 16}

数字相乘:

2 \cdot 16 = 32

= \frac{2 - 2 17}{- 32}

使用分式法则:

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

2 - 217 = - (217 - 2)

= \frac{2 17 - 2}{32}

分解

217 - 2 : 2 (17 - 1)

217 - 2

改写为

= 217 - 2 \cdot 1

因式分解出通项

2

= 2 (17 - 1)

= \frac{2 ( 17 - 1 )}{32}

约分:

2

= \frac{17 - 1}{16}

二次方程组的解是:

u = - \frac{1 + 17}{16}, u = \frac{17 - 1}{16}

u = sin (x)

代回

sin (x) = - \frac{1 + 17}{16}, sin (x) = \frac{17 - 1}{16}

sin (x) = - \frac{1 + 17}{16}, sin (x) = \frac{17 - 1}{16}

sin (x) = - \frac{1 + 17}{16} : x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn

sin (x) = - \frac{1 + 17}{16}

使用反三角函数性质

sin (x) = - \frac{1 + 17}{16}

sin (x) = - \frac{1 + 17}{16}

的通解

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

x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn

x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn

sin (x) = \frac{17 - 1}{16} : x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

sin (x) = \frac{17 - 1}{16}

使用反三角函数性质

sin (x) = \frac{17 - 1}{16}

sin (x) = \frac{17 - 1}{16}

的通解

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

x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

合并所有解

x = arcsin (- \frac{1 + 17}{16}) + 2 πn, x = π + arcsin (\frac{1 + 17}{16}) + 2 πn, x = arcsin (\frac{17 - 1}{16}) + 2 πn, x = π - arcsin (\frac{17 - 1}{16}) + 2 πn

以小数形式表示解

x = - 0.32593 \dots + 2 πn, x = π + 0.32593 \dots + 2 πn, x = 0.19645 \dots + 2 πn, x = π - 0.19645 \dots + 2 πn

作图

流行的例子

cos(θ)=0.5332,θ

cos(5a+40)= 3/5

2cos(x^2)-sqrt(2)=0

solvefor x,sin(x)=-sqrt(2)*cos(x)