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受欢迎的三角函数 >

cos(x)=2-3sin(x)

解答

cos (x) = 2 - 3 sin (x)

解答

x = 0.36296 \dots + 2 πn, x = π - 1.00646 \dots + 2 πn

+1

度数

x = 20.79657 \dots^{\circ} + 36 0^{\circ} n, x = 122.33353 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos (x) = 2 - 3 sin (x)

两边进行平方

cos^{2} (x) = (2 - 3 sin (x))^{2}

两边减去

(2 - 3 sin (x))^{2}

cos^{2} (x) - 4 + 12 sin (x) - 9 sin^{2} (x) = 0

使用三角恒等式改写

- 4 + cos^{2} (x) + 12 sin (x) - 9 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)

= - 4 + 1 - sin^{2} (x) + 12 sin (x) - 9 sin^{2} (x)

化简

- 4 + 1 - sin^{2} (x) + 12 sin (x) - 9 sin^{2} (x) : 12 sin (x) - 10 sin^{2} (x) - 3

- 4 + 1 - sin^{2} (x) + 12 sin (x) - 9 sin^{2} (x)

对同类项分组

= - sin^{2} (x) + 12 sin (x) - 9 sin^{2} (x) - 4 + 1

同类项相加:

- sin^{2} (x) - 9 sin^{2} (x) = - 10 sin^{2} (x)

= - 10 sin^{2} (x) + 12 sin (x) - 4 + 1

数字相加/相减:

- 4 + 1 = - 3

= 12 sin (x) - 10 sin^{2} (x) - 3

= 12 sin (x) - 10 sin^{2} (x) - 3

- 3 - 10 sin^{2} (x) + 12 sin (x) = 0

用替代法求解

- 3 - 10 sin^{2} (x) + 12 sin (x) = 0

令

: sin (x) = u

- 3 - 10 u^{2} + 12 u = 0

- 3 - 10 u^{2} + 12 u = 0 : u = \frac{6 - 6}{10}, u = \frac{6 + 6}{10}

- 3 - 10 u^{2} + 12 u = 0

改写成标准形式

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

- 10 u^{2} + 12 u - 3 = 0

使用求根公式求解

- 10 u^{2} + 12 u - 3 = 0

二次方程求根公式:

若

a = - 10, b = 12, c = - 3

u_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 ( - 10 ) ( - 3 )}{2 ( - 10 )}

u_{1, 2} = \frac{- 12 \pm 1 2 ^{2} - 4 ( - 10 ) ( - 3 )}{2 ( - 10 )}

1 2^{2} - 4 (- 10) (- 3) = 26

1 2^{2} - 4 (- 10) (- 3)

使用法则

- (- a) = a

= 1 2^{2} - 4 \cdot 10 \cdot 3

数字相乘:

4 \cdot 10 \cdot 3 = 120

= 1 2^{2} - 120

1 2^{2} = 144

= 144 - 120

数字相减:

144 - 120 = 24

= 24

24

质因数分解:

2^{3} \cdot 3

24

24

除以

224 = 12 \cdot 2

= 2 \cdot 12

12

除以

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \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

= 2^{3} \cdot 3

= 2^{3} \cdot 3

使用指数法则:

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

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

使用根式运算法则:

= 2^{2} 2 \cdot 3

使用根式运算法则:

2^{2} = 2

= 2 2 \cdot 3

整理后得

= 26

u_{1, 2} = \frac{- 12 \pm 2 6}{2 ( - 10 )}

将解分隔开

u_{1} = \frac{- 12 + 2 6}{2 ( - 10 )}, u_{2} = \frac{- 12 - 2 6}{2 ( - 10 )}

u = \frac{- 12 + 2 6}{2 ( - 10 )} : \frac{6 - 6}{10}

\frac{- 12 + 2 6}{2 ( - 10 )}

去除括号:

(- a) = - a

= \frac{- 12 + 2 6}{- 2 \cdot 10}

数字相乘:

2 \cdot 10 = 20

= \frac{- 12 + 2 6}{- 20}

使用分式法则:

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

- 12 + 26 = - (12 - 26)

= \frac{12 - 2 6}{20}

分解

12 - 26 : 2 (6 - 6)

12 - 26

改写为

= 2 \cdot 6 - 26

因式分解出通项

2

= 2 (6 - 6)

= \frac{2 ( 6 - 6 )}{20}

约分:

2

= \frac{6 - 6}{10}

u = \frac{- 12 - 2 6}{2 ( - 10 )} : \frac{6 + 6}{10}

\frac{- 12 - 2 6}{2 ( - 10 )}

去除括号:

(- a) = - a

= \frac{- 12 - 2 6}{- 2 \cdot 10}

数字相乘:

2 \cdot 10 = 20

= \frac{- 12 - 2 6}{- 20}

使用分式法则:

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

- 12 - 26 = - (12 + 26)

= \frac{12 + 2 6}{20}

分解

12 + 26 : 2 (6 + 6)

12 + 26

改写为

= 2 \cdot 6 + 26

因式分解出通项

2

= 2 (6 + 6)

= \frac{2 ( 6 + 6 )}{20}

约分:

2

= \frac{6 + 6}{10}

二次方程组的解是:

u = \frac{6 - 6}{10}, u = \frac{6 + 6}{10}

u = sin (x)

代回

sin (x) = \frac{6 - 6}{10}, sin (x) = \frac{6 + 6}{10}

sin (x) = \frac{6 - 6}{10}, sin (x) = \frac{6 + 6}{10}

sin (x) = \frac{6 - 6}{10} : x = arcsin (\frac{6 - 6}{10}) + 2 πn, x = π - arcsin (\frac{6 - 6}{10}) + 2 πn

sin (x) = \frac{6 - 6}{10}

使用反三角函数性质

sin (x) = \frac{6 - 6}{10}

sin (x) = \frac{6 - 6}{10}

的通解

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

x = arcsin (\frac{6 - 6}{10}) + 2 πn, x = π - arcsin (\frac{6 - 6}{10}) + 2 πn

x = arcsin (\frac{6 - 6}{10}) + 2 πn, x = π - arcsin (\frac{6 - 6}{10}) + 2 πn

sin (x) = \frac{6 + 6}{10} : x = arcsin (\frac{6 + 6}{10}) + 2 πn, x = π - arcsin (\frac{6 + 6}{10}) + 2 πn

sin (x) = \frac{6 + 6}{10}

使用反三角函数性质

sin (x) = \frac{6 + 6}{10}

sin (x) = \frac{6 + 6}{10}

的通解

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

x = arcsin (\frac{6 + 6}{10}) + 2 πn, x = π - arcsin (\frac{6 + 6}{10}) + 2 πn

x = arcsin (\frac{6 + 6}{10}) + 2 πn, x = π - arcsin (\frac{6 + 6}{10}) + 2 πn

合并所有解

x = arcsin (\frac{6 - 6}{10}) + 2 πn, x = π - arcsin (\frac{6 - 6}{10}) + 2 πn, x = arcsin (\frac{6 + 6}{10}) + 2 πn, x = π - arcsin (\frac{6 + 6}{10}) + 2 πn

将解代入原方程进行验证

将它们代入

cos (x) = 2 - 3 sin (x)

检验解是否符合

去除与方程不符的解。

检验

arcsin (\frac{6 - 6}{10}) + 2 πn

的解:

真

arcsin (\frac{6 - 6}{10}) + 2 πn

代入

n = 1

arcsin (\frac{6 - 6}{10}) + 2 π 1

对于

cos (x) = 2 - 3 sin (x)

代入

x = arcsin (\frac{6 - 6}{10}) + 2 π 1

cos (arcsin (\frac{6 - 6}{10}) + 2 π 1) = 2 - 3 sin (arcsin (\frac{6 - 6}{10}) + 2 π 1)

整理后得

0.93484 \dots = 0.93484 \dots

\Rightarrow 真

检验

π - arcsin (\frac{6 - 6}{10}) + 2 πn

的解:

假

π - arcsin (\frac{6 - 6}{10}) + 2 πn

代入

n = 1

π - arcsin (\frac{6 - 6}{10}) + 2 π 1

对于

cos (x) = 2 - 3 sin (x)

代入

x = π - arcsin (\frac{6 - 6}{10}) + 2 π 1

cos (π - arcsin (\frac{6 - 6}{10}) + 2 π 1) = 2 - 3 sin (π - arcsin (\frac{6 - 6}{10}) + 2 π 1)

整理后得

- 0.93484 \dots = 0.93484 \dots

\Rightarrow 假

检验

arcsin (\frac{6 + 6}{10}) + 2 πn

的解:

假

arcsin (\frac{6 + 6}{10}) + 2 πn

代入

n = 1

arcsin (\frac{6 + 6}{10}) + 2 π 1

对于

cos (x) = 2 - 3 sin (x)

代入

x = arcsin (\frac{6 + 6}{10}) + 2 π 1

cos (arcsin (\frac{6 + 6}{10}) + 2 π 1) = 2 - 3 sin (arcsin (\frac{6 + 6}{10}) + 2 π 1)

整理后得

0.53484 \dots = - 0.53484 \dots

\Rightarrow 假

检验

π - arcsin (\frac{6 + 6}{10}) + 2 πn

的解:

真

π - arcsin (\frac{6 + 6}{10}) + 2 πn

代入

n = 1

π - arcsin (\frac{6 + 6}{10}) + 2 π 1

对于

cos (x) = 2 - 3 sin (x)

代入

x = π - arcsin (\frac{6 + 6}{10}) + 2 π 1

cos (π - arcsin (\frac{6 + 6}{10}) + 2 π 1) = 2 - 3 sin (π - arcsin (\frac{6 + 6}{10}) + 2 π 1)

整理后得

- 0.53484 \dots = - 0.53484 \dots

\Rightarrow 真

x = arcsin (\frac{6 - 6}{10}) + 2 πn, x = π - arcsin (\frac{6 + 6}{10}) + 2 πn

以小数形式表示解

x = 0.36296 \dots + 2 πn, x = π - 1.00646 \dots + 2 πn

作图

流行的例子

2cos^2(x)-1-cos(x)=0,0<= x<= 2pi

tan^2(x)-tan(x)=2

2sin^2(x/2)-cos(x/2)=0