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受欢迎的三角函数 >

4sin^2(x)+(2sqrt(3)+2)sin(x)+sqrt(3)=0

解答

4 sin^{2} (x) + (23 + 2) sin (x) + 3 = 0

解答

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

+1

度数

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sin^{2} (x) + (23 + 2) sin (x) + 3 = 0

用替代法求解

4 sin^{2} (x) + (23 + 2) sin (x) + 3 = 0

令

: sin (x) = u

4 u^{2} + (23 + 2) u + 3 = 0

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

4 u^{2} + (23 + 2) u + 3 = 0

使用求根公式求解

4 u^{2} + (23 + 2) u + 3 = 0

二次方程求根公式:

若

a = 4, b = 23 + 2, c = 3

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

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

(23 + 2)^{2} - 4 \cdot 43 = 23 - 2

(23 + 2)^{2} - 4 \cdot 43

数字相乘:

4 \cdot 4 = 16

= (23 + 2)^{2} - 163

乘开

(23 + 2)^{2} - 163 : 16 - 83

(23 + 2)^{2} - 163

(23 + 2)^{2} : 16 + 83

使用完全平方公式:

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

a = 23, b = 2

= (23)^{2} + 2 \cdot 23 \cdot 2 + 2^{2}

化简

(23)^{2} + 2 \cdot 23 \cdot 2 + 2^{2} : 16 + 83

(23)^{2} + 2 \cdot 23 \cdot 2 + 2^{2}

(23)^{2} = 12

(23)^{2}

使用指数法则:

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

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

(3)^{2} : 3

使用根式运算法则:

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

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

使用指数法则:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

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

约分:

2

= 1

= 3

= 2^{2} \cdot 3

2^{2} = 4

= 4 \cdot 3

数字相乘:

4 \cdot 3 = 12

= 12

2 \cdot 23 \cdot 2 = 83

2 \cdot 23 \cdot 2

数字相乘:

2 \cdot 2 \cdot 2 = 8

= 83

2^{2} = 4

2^{2}

2^{2} = 4

= 4

= 12 + 83 + 4

数字相加:

12 + 4 = 16

= 16 + 83

= 16 + 83

= 16 + 83 - 163

同类项相加:

83 - 163 = - 83

= 16 - 83

= 16 - 83

= 12 - 83 + 4

= 4 \cdot 3 - 83 + 4

= (4)^{2} (3)^{2} - 83 + (4)^{2}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= 2^{2} (3)^{2} - 83 + 2^{2}

2 \cdot 23 \cdot 2 = 83

2 \cdot 23 \cdot 2

数字相乘:

2 \cdot 2 \cdot 2 = 8

= 83

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

使用完全平方公式:

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

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

= (23 - 2)^{2}

使用根式运算法则:

n a^{n} = a

(23 - 2)^{2} = 23 - 2

= 23 - 2

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

将解分隔开

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

u = \frac{- ( 2 3 + 2 ) + 2 3 - 2}{2 \cdot 4} : - \frac{1}{2}

\frac{- ( 2 3 + 2 ) + 2 3 - 2}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- ( 2 3 + 2 ) + 2 3 - 2}{8}

乘开

- (23 + 2) + 23 - 2 : - 4

- (23 + 2) + 23 - 2

- (23 + 2) : - 23 - 2

- (23 + 2)

打开括号

= - (23) - (2)

使用加减运算法则

+ (- a) = - a

= - 23 - 2

= - 23 - 2 + 23 - 2

化简

- 23 - 2 + 23 - 2 : - 4

- 23 - 2 + 23 - 2

同类项相加:

- 23 + 23 = 0

= - 2 - 2

数字相减:

- 2 - 2 = - 4

= - 4

= - 4

= \frac{- 4}{8}

使用分式法则:

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

= - \frac{4}{8}

约分:

4

= - \frac{1}{2}

u = \frac{- ( 2 3 + 2 ) - ( 2 3 - 2 )}{2 \cdot 4} : - \frac{3}{2}

\frac{- ( 2 3 + 2 ) - ( 2 3 - 2 )}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- ( 2 3 + 2 ) - ( 2 3 - 2 )}{8}

乘开

- (23 + 2) - (23 - 2) : - 43

- (23 + 2) - (23 - 2)

- (23 + 2) : - 23 - 2

- (23 + 2)

打开括号

= - (23) - (2)

使用加减运算法则

+ (- a) = - a

= - 23 - 2

= - 23 - 2 - (23 - 2)

- (23 - 2) : - 23 + 2

- (23 - 2)

打开括号

= - (23) - (- 2)

使用加减运算法则

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

= - 23 + 2

= - 23 - 2 - 23 + 2

化简

- 23 - 2 - 23 + 2 : - 43

- 23 - 2 - 23 + 2

同类项相加:

- 23 - 23 = - 43

= - 43 - 2 + 2

- 2 + 2 = 0

= - 43

= - 43

= \frac{- 4 3}{8}

使用分式法则:

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

= - \frac{4 3}{8}

约分:

4

= - \frac{3}{2}

二次方程组的解是:

u = - \frac{1}{2}, u = - \frac{3}{2}

u = sin (x)

代回

sin (x) = - \frac{1}{2}, sin (x) = - \frac{3}{2}

sin (x) = - \frac{1}{2}, sin (x) = - \frac{3}{2}

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

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

sin (x) = - \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}

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

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

sin (x) = - \frac{3}{2} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

合并所有解

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

作图

流行的例子

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

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

3csc(x)=2csc(x)+sqrt(2)

3 csc (x) = 2 csc (x) + 2

sec(x)=tan(x)+cot(x)

sec (x) = tan (x) + cot (x)

solvefor x,cos(x)= 1/3

so l v e f or x, cos (x) = \frac{1}{3}

cos (θ) = - 0.02