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受欢迎的三角函数 >

3-3sin(t)=3sqrt(3)cos(t)

解答

3 - 3 sin (t) = 33 cos (t)

解答

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

+1

度数

t = 9 0^{\circ} + 36 0^{\circ} n, t = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

3 - 3 sin (t) = 33 cos (t)

两边进行平方

(3 - 3 sin (t))^{2} = (33 cos (t))^{2}

两边减去

(33 cos (t))^{2}

(3 - 3 sin (t))^{2} - 27 cos^{2} (t) = 0

使用三角恒等式改写

(3 - 3 sin (t))^{2} - 27 cos^{2} (t)

使用毕达哥拉斯恒等式:

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

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

= (3 - 3 sin (t))^{2} - 27 (1 - sin^{2} (t))

化简

(3 - 3 sin (t))^{2} - 27 (1 - sin^{2} (t)) : 36 sin^{2} (t) - 18 sin (t) - 18

(3 - 3 sin (t))^{2} - 27 (1 - sin^{2} (t))

(3 - 3 sin (t))^{2} : 9 - 18 sin (t) + 9 sin^{2} (t)

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 3, b = 3 sin (t)

= 3^{2} - 2 \cdot 3 \cdot 3 sin (t) + (3 sin (t))^{2}

化简

3^{2} - 2 \cdot 3 \cdot 3 sin (t) + (3 sin (t))^{2} : 9 - 18 sin (t) + 9 sin^{2} (t)

3^{2} - 2 \cdot 3 \cdot 3 sin (t) + (3 sin (t))^{2}

3^{2} = 9

3^{2}

3^{2} = 9

= 9

2 \cdot 3 \cdot 3 sin (t) = 18 sin (t)

2 \cdot 3 \cdot 3 sin (t)

数字相乘:

2 \cdot 3 \cdot 3 = 18

= 18 sin (t)

(3 sin (t))^{2} = 9 sin^{2} (t)

(3 sin (t))^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 3^{2} sin^{2} (t)

3^{2} = 9

= 9 sin^{2} (t)

= 9 - 18 sin (t) + 9 sin^{2} (t)

= 9 - 18 sin (t) + 9 sin^{2} (t)

= 9 - 18 sin (t) + 9 sin^{2} (t) - 27 (1 - sin^{2} (t))

乘开

- 27 (1 - sin^{2} (t)) : - 27 + 27 sin^{2} (t)

- 27 (1 - sin^{2} (t))

使用分配律:

a (b - c) = ab - a c

a = - 27, b = 1, c = sin^{2} (t)

= - 27 \cdot 1 - (- 27) sin^{2} (t)

使用加减运算法则

- (- a) = a

= - 27 \cdot 1 + 27 sin^{2} (t)

数字相乘:

27 \cdot 1 = 27

= - 27 + 27 sin^{2} (t)

= 9 - 18 sin (t) + 9 sin^{2} (t) - 27 + 27 sin^{2} (t)

化简

9 - 18 sin (t) + 9 sin^{2} (t) - 27 + 27 sin^{2} (t) : 36 sin^{2} (t) - 18 sin (t) - 18

9 - 18 sin (t) + 9 sin^{2} (t) - 27 + 27 sin^{2} (t)

对同类项分组

= - 18 sin (t) + 9 sin^{2} (t) + 27 sin^{2} (t) + 9 - 27

同类项相加:

9 sin^{2} (t) + 27 sin^{2} (t) = 36 sin^{2} (t)

= - 18 sin (t) + 36 sin^{2} (t) + 9 - 27

数字相加/相减:

9 - 27 = - 18

= 36 sin^{2} (t) - 18 sin (t) - 18

= 36 sin^{2} (t) - 18 sin (t) - 18

= 36 sin^{2} (t) - 18 sin (t) - 18

- 18 - 18 sin (t) + 36 sin^{2} (t) = 0

用替代法求解

- 18 - 18 sin (t) + 36 sin^{2} (t) = 0

令

: sin (t) = u

- 18 - 18 u + 36 u^{2} = 0

- 18 - 18 u + 36 u^{2} = 0 : u = 1, u = - \frac{1}{2}

- 18 - 18 u + 36 u^{2} = 0

改写成标准形式

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

36 u^{2} - 18 u - 18 = 0

使用求根公式求解

36 u^{2} - 18 u - 18 = 0

二次方程求根公式:

若

a = 36, b = - 18, c = - 18

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 36 ( - 18 )}{2 \cdot 36}

u_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 36 ( - 18 )}{2 \cdot 36}

(- 18)^{2} - 4 \cdot 36 (- 18) = 54

(- 18)^{2} - 4 \cdot 36 (- 18)

使用法则

- (- a) = a

= (- 18)^{2} + 4 \cdot 36 \cdot 18

使用指数法则:

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

若

n

是偶数

(- 18)^{2} = 1 8^{2}

= 1 8^{2} + 4 \cdot 36 \cdot 18

数字相乘:

4 \cdot 36 \cdot 18 = 2592

= 1 8^{2} + 2592

1 8^{2} = 324

= 324 + 2592

数字相加:

324 + 2592 = 2916

= 2916

因式分解数字:

2916 = 5 4^{2}

= 5 4^{2}

使用根式运算法则:

n a^{n} = a

5 4^{2} = 54

= 54

u_{1, 2} = \frac{- ( - 18 ) \pm 54}{2 \cdot 36}

将解分隔开

u_{1} = \frac{- ( - 18 ) + 54}{2 \cdot 36}, u_{2} = \frac{- ( - 18 ) - 54}{2 \cdot 36}

u = \frac{- ( - 18 ) + 54}{2 \cdot 36} : 1

\frac{- ( - 18 ) + 54}{2 \cdot 36}

使用法则

- (- a) = a

= \frac{18 + 54}{2 \cdot 36}

数字相加:

18 + 54 = 72

= \frac{72}{2 \cdot 36}

数字相乘:

2 \cdot 36 = 72

= \frac{72}{72}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- ( - 18 ) - 54}{2 \cdot 36} : - \frac{1}{2}

\frac{- ( - 18 ) - 54}{2 \cdot 36}

使用法则

- (- a) = a

= \frac{18 - 54}{2 \cdot 36}

数字相减:

18 - 54 = - 36

= \frac{- 36}{2 \cdot 36}

数字相乘:

2 \cdot 36 = 72

= \frac{- 36}{72}

使用分式法则:

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

= - \frac{36}{72}

约分:

36

= - \frac{1}{2}

二次方程组的解是:

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

u = sin (t)

代回

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

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

sin (t) = 1 : t = \frac{π}{2} + 2 πn

sin (t) = 1

sin (t) = 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}

t = \frac{π}{2} + 2 πn

t = \frac{π}{2} + 2 πn

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

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

sin (t) = - \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}

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

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

合并所有解

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

将解代入原方程进行验证

将它们代入

3 - 3 sin (t) = 33 cos (t)

检验解是否符合

去除与方程不符的解。

检验

\frac{π}{2} + 2 πn

的解:

真

\frac{π}{2} + 2 πn

代入

n = 1

\frac{π}{2} + 2 π 1

对于

3 - 3 sin (t) = 33 cos (t)

代入

t = \frac{π}{2} + 2 π 1

3 - 3 sin (\frac{π}{2} + 2 π 1) = 33 cos (\frac{π}{2} + 2 π 1)

整理后得

0 = 0

\Rightarrow 真

检验

\frac{7 π}{6} + 2 πn

的解:

假

\frac{7 π}{6} + 2 πn

代入

n = 1

\frac{7 π}{6} + 2 π 1

对于

3 - 3 sin (t) = 33 cos (t)

代入

t = \frac{7 π}{6} + 2 π 1

3 - 3 sin (\frac{7 π}{6} + 2 π 1) = 33 cos (\frac{7 π}{6} + 2 π 1)

整理后得

4.5 = - 4.5

\Rightarrow 假

检验

\frac{11 π}{6} + 2 πn

的解:

真

\frac{11 π}{6} + 2 πn

代入

n = 1

\frac{11 π}{6} + 2 π 1

对于

3 - 3 sin (t) = 33 cos (t)

代入

t = \frac{11 π}{6} + 2 π 1

3 - 3 sin (\frac{11 π}{6} + 2 π 1) = 33 cos (\frac{11 π}{6} + 2 π 1)

整理后得

4.5 = 4.5

\Rightarrow 真

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

作图

流行的例子

sin (θ) = \frac{3}{10}

30arctan(x)=5pi

30 arctan (x) = 5 π

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cosh (2 x) - 10 cosh (x) + 13 = 0

sqrt(3)sin(x)-sin(2x)=0

3 sin (x) - sin (2 x) = 0

7cos^2(x)+14cos(x)+7=0

7 cos^{2} (x) + 14 cos (x) + 7 = 0