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受欢迎的三角函数 >

2sin^2(x)=4sin(x)+6,(0,2pi)

解答

2 sin^{2} (x) = 4 sin (x) + 6, (0, 2 π)

解答

x = \frac{3 π}{2}

+1

度数

x = 27 0^{\circ}

求解步骤

2 sin^{2} (x) = 4 sin (x) + 6, (0, 2 π)

用替代法求解

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

令

: sin (x) = u

2 u^{2} = 4 u + 6

2 u^{2} = 4 u + 6 : u = 3, u = - 1

2 u^{2} = 4 u + 6

将

6

para o lado esquerdo

2 u^{2} = 4 u + 6

两边减去

6

2 u^{2} - 6 = 4 u + 6 - 6

化简

2 u^{2} - 6 = 4 u

2 u^{2} - 6 = 4 u

将

4 u

para o lado esquerdo

2 u^{2} - 6 = 4 u

两边减去

4 u

2 u^{2} - 6 - 4 u = 4 u - 4 u

化简

2 u^{2} - 6 - 4 u = 0

2 u^{2} - 6 - 4 u = 0

改写成标准形式

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

2 u^{2} - 4 u - 6 = 0

使用求根公式求解

2 u^{2} - 4 u - 6 = 0

二次方程求根公式:

若

a = 2, b = - 4, c = - 6

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

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

(- 4)^{2} - 4 \cdot 2 (- 6) = 8

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

(- 4)^{2} = 4^{2}

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

数字相乘:

4 \cdot 2 \cdot 6 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数字相加:

16 + 48 = 64

= 64

因式分解数字:

64 = 8^{2}

= 8^{2}

使用根式运算法则:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 4 ) \pm 8}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 4 ) + 8}{2 \cdot 2}, u_{2} = \frac{- ( - 4 ) - 8}{2 \cdot 2}

u = \frac{- ( - 4 ) + 8}{2 \cdot 2} : 3

\frac{- ( - 4 ) + 8}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{4 + 8}{2 \cdot 2}

数字相加:

4 + 8 = 12

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

数字相乘:

2 \cdot 2 = 4

= \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 4 ) - 8}{2 \cdot 2} : - 1

\frac{- ( - 4 ) - 8}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{4 - 8}{2 \cdot 2}

数字相减:

4 - 8 = - 4

= \frac{- 4}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{4}

使用分式法则:

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

= - \frac{4}{4}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = 3, u = - 1

u = sin (x)

代回

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

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

sin (x) = 3, (0, 2 π) :

无解

sin (x) = 3, (0, 2 π)

- 1 \leq sin (x) \leq 1

无解

sin (x) = - 1, (0, 2 π) : x = \frac{3 π}{2}

sin (x) = - 1, (0, 2 π)

sin (x) = - 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}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

在

(0, 2 π)

范围内的解

x = \frac{3 π}{2}

合并所有解

x = \frac{3 π}{2}

作图

流行的例子

2 cos (3 x + 2 0^{\circ}) = 2

cot^2(3x)-2cot(3x)+1=0

cot^{2} (3 x) - 2 cot (3 x) + 1 = 0

-(tanh(nx))/(cosh(nx))=0

- \frac{tanh ( n x )}{cosh ( n x )} = 0

sin (φ) = 0

sin^2(x)=1-tan^2(x),0<= x<= pi/2

sin^{2} (x) = 1 - tan^{2} (x), 0 \leq x \leq \frac{π}{2}