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受欢迎的三角函数 >

2sqrt(3)cos^2(x)=sin(x)

解答

23 cos^{2} (x) = sin (x)

解答

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

求解步骤

23 cos^{2} (x) = sin (x)

两边减去

sin (x)

23 cos^{2} (x) - sin (x) = 0

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

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

- sin (x) + (1 - sin^{2} (x)) \cdot 23 = 0

用替代法求解

- sin (x) + (1 - sin^{2} (x)) \cdot 23 = 0

令

: sin (x) = u

- u + (1 - u^{2}) \cdot 23 = 0

- u + (1 - u^{2}) \cdot 23 = 0 : u = - \frac{2 3}{3}, u = \frac{3}{2}

- u + (1 - u^{2}) \cdot 23 = 0

展开

- u + (1 - u^{2}) \cdot 23 : - u + 23 - 23 u^{2}

- u + (1 - u^{2}) \cdot 23

= - u + 23 (1 - u^{2})

乘开

23 (1 - u^{2}) : 23 - 23 u^{2}

23 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 23, b = 1, c = u^{2}

= 23 \cdot 1 - 23 u^{2}

= 2 \cdot 1 \cdot 3 - 23 u^{2}

数字相乘:

2 \cdot 1 = 2

= 23 - 23 u^{2}

= - u + 23 - 23 u^{2}

- u + 23 - 23 u^{2} = 0

改写成标准形式

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

- 23 u^{2} - u + 23 = 0

使用求根公式求解

- 23 u^{2} - u + 23 = 0

二次方程求根公式:

若

a = - 23, b = - 1, c = 23

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

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

(- 1)^{2} - 4 (- 23) \cdot 23 = 7

(- 1)^{2} - 4 (- 23) \cdot 23

使用法则

- (- a) = a

= (- 1)^{2} + 4 \cdot 23 \cdot 23

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 23 \cdot 23 = 48

4 \cdot 23 \cdot 23

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 1633

使用根式运算法则:

a a = a

33 = 3

= 16 \cdot 3

数字相乘:

16 \cdot 3 = 48

= 48

= 1 + 48

数字相加:

1 + 48 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 ( - 2 3 )}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 7}{2 ( - 2 3 )}, u_{2} = \frac{- ( - 1 ) - 7}{2 ( - 2 3 )}

u = \frac{- ( - 1 ) + 7}{2 ( - 2 3 )} : - \frac{2 3}{3}

\frac{- ( - 1 ) + 7}{2 ( - 2 3 )}

去除括号:

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

= \frac{1 + 7}{- 2 \cdot 2 3}

数字相加:

1 + 7 = 8

= \frac{8}{- 2 \cdot 2 3}

数字相乘:

2 \cdot 2 = 4

= \frac{8}{- 4 3}

使用分式法则:

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

= - \frac{8}{4 3}

数字相除:

\frac{8}{4} = 2

= - \frac{2}{3}

- \frac{2}{3}

有理化:

- \frac{2 3}{3}

- \frac{2}{3}

乘以共轭根式

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{- ( - 1 ) - 7}{2 ( - 2 3 )} : \frac{3}{2}

\frac{- ( - 1 ) - 7}{2 ( - 2 3 )}

去除括号:

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

= \frac{1 - 7}{- 2 \cdot 2 3}

数字相减:

1 - 7 = - 6

= \frac{- 6}{- 2 \cdot 2 3}

数字相乘:

2 \cdot 2 = 4

= \frac{- 6}{- 4 3}

使用分式法则:

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

= \frac{6}{4 3}

约分:

2

= \frac{3}{2 3}

使用根式运算法则:

n a = a^{\frac{1}{n}}

3 = 3^{\frac{1}{2}}

= \frac{3}{2 \cdot 3 ^{\frac{1}{2}}}

使用指数法则:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= \frac{3 ^{1 - \frac{1}{2}}}{2}

数字相减:

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

= \frac{3 ^{\frac{1}{2}}}{2}

使用根式运算法则:

a^{\frac{1}{n}} = n a

3^{\frac{1}{2}} = 3

= \frac{3}{2}

二次方程组的解是:

u = - \frac{2 3}{3}, u = \frac{3}{2}

u = sin (x)

代回

sin (x) = - \frac{2 3}{3}, sin (x) = \frac{3}{2}

sin (x) = - \frac{2 3}{3}, sin (x) = \frac{3}{2}

sin (x) = - \frac{2 3}{3} :

无解

sin (x) = - \frac{2 3}{3}

- 1 \leq sin (x) \leq 1

无解

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

sin (x) = \frac{3}{2}

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

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

合并所有解

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

作图

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