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受欢迎的三角函数 >

cos(x)+3sin(x)=1

解答

cos (x) + 3 sin (x) = 1

解答

x = 2 πn, x = π - 0.64350 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 143.13010 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos (x) + 3 sin (x) = 1

两边减去

3 sin (x)

cos (x) = 1 - 3 sin (x)

两边进行平方

cos^{2} (x) = (1 - 3 sin (x))^{2}

两边减去

(1 - 3 sin (x))^{2}

cos^{2} (x) - 1 + 6 sin (x) - 9 sin^{2} (x) = 0

使用三角恒等式改写

- 1 + cos^{2} (x) + 6 sin (x) - 9 sin^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= 6 sin (x) - 9 sin^{2} (x) - sin^{2} (x)

化简

= 6 sin (x) - 10 sin^{2} (x)

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

用替代法求解

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

令

: sin (x) = u

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

- 10 u^{2} + 6 u = 0 : u = 0, u = \frac{3}{5}

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 10, b = 6, c = 0

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 10 ) \cdot 0}{2 ( - 10 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 10 ) \cdot 0}{2 ( - 10 )}

6^{2} - 4 (- 10) \cdot 0 = 6

6^{2} - 4 (- 10) \cdot 0

使用法则

- (- a) = a

= 6^{2} + 4 \cdot 10 \cdot 0

使用法则

0 \cdot a = 0

= 6^{2} + 0

6^{2} + 0 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 6

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

将解分隔开

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

u = \frac{- 6 + 6}{2 ( - 10 )} : 0

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

去除括号:

(- a) = - a

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

数字相加/相减:

- 6 + 6 = 0

= \frac{0}{- 2 \cdot 10}

数字相乘:

2 \cdot 10 = 20

= \frac{0}{- 20}

使用分式法则:

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

= - \frac{0}{20}

使用法则

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 6 - 6}{2 ( - 10 )} : \frac{3}{5}

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

去除括号:

(- a) = - a

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

数字相减:

- 6 - 6 = - 12

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

数字相乘:

2 \cdot 10 = 20

= \frac{- 12}{- 20}

使用分式法则:

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

= \frac{12}{20}

约分:

4

= \frac{3}{5}

二次方程组的解是:

u = 0, u = \frac{3}{5}

u = sin (x)

代回

sin (x) = 0, sin (x) = \frac{3}{5}

sin (x) = 0, sin (x) = \frac{3}{5}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

sin (x) = \frac{3}{5}

使用反三角函数性质

sin (x) = \frac{3}{5}

sin (x) = \frac{3}{5}

的通解

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

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

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

合并所有解

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

将解代入原方程进行验证

将它们代入

cos (x) + 3 sin (x) = 1

检验解是否符合

去除与方程不符的解。

检验

2 πn

的解:

真

2 πn

代入

n = 1

2 π 1

对于

cos (x) + 3 sin (x) = 1

代入

x = 2 π 1

cos (2 π 1) + 3 sin (2 π 1) = 1

整理后得

1 = 1

\Rightarrow 真

检验

π + 2 πn

的解:

假

π + 2 πn

代入

n = 1

π + 2 π 1

对于

cos (x) + 3 sin (x) = 1

代入

x = π + 2 π 1

cos (π + 2 π 1) + 3 sin (π + 2 π 1) = 1

整理后得

- 1 = 1

\Rightarrow 假

检验

arcsin (\frac{3}{5}) + 2 πn

的解:

假

arcsin (\frac{3}{5}) + 2 πn

代入

n = 1

arcsin (\frac{3}{5}) + 2 π 1

对于

cos (x) + 3 sin (x) = 1

代入

x = arcsin (\frac{3}{5}) + 2 π 1

cos (arcsin (\frac{3}{5}) + 2 π 1) + 3 sin (arcsin (\frac{3}{5}) + 2 π 1) = 1

整理后得

2.6 = 1

\Rightarrow 假

检验

π - arcsin (\frac{3}{5}) + 2 πn

的解:

真

π - arcsin (\frac{3}{5}) + 2 πn

代入

n = 1

π - arcsin (\frac{3}{5}) + 2 π 1

对于

cos (x) + 3 sin (x) = 1

代入

x = π - arcsin (\frac{3}{5}) + 2 π 1

cos (π - arcsin (\frac{3}{5}) + 2 π 1) + 3 sin (π - arcsin (\frac{3}{5}) + 2 π 1) = 1

整理后得

1 = 1

\Rightarrow 真

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

以小数形式表示解

x = 2 πn, x = π - 0.64350 \dots + 2 πn

作图

流行的例子

2sin^2(x)=4sin(x)+6

2 sin^{2} (x) = 4 sin (x) + 6

cos (x) = sin (1 5^{\circ})

3 = 3 tan (x)

2 cot (x) + 1 = - 1

sin(θ)= 3/5 ,cos(θ)

sin (θ) = \frac{3}{5}, cos (θ)