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受欢迎的三角函数 >

4sin^4(x)+3sin^2(x)-1=0

解答

4 sin^{4} (x) + 3 sin^{2} (x) - 1 = 0

解答

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

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

4 sin^{4} (x) + 3 sin^{2} (x) - 1 = 0

用替代法求解

4 sin^{4} (x) + 3 sin^{2} (x) - 1 = 0

令

: sin (x) = u

4 u^{4} + 3 u^{2} - 1 = 0

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

4 u^{4} + 3 u^{2} - 1 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

4 v^{2} + 3 v - 1 = 0

解

4 v^{2} + 3 v - 1 = 0 : v = \frac{1}{4}, v = - 1

4 v^{2} + 3 v - 1 = 0

使用求根公式求解

4 v^{2} + 3 v - 1 = 0

二次方程求根公式:

若

a = 4, b = 3, c = - 1

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

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

3^{2} - 4 \cdot 4 (- 1) = 5

3^{2} - 4 \cdot 4 (- 1)

使用法则

- (- a) = a

= 3^{2} + 4 \cdot 4 \cdot 1

数字相乘:

4 \cdot 4 \cdot 1 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

v_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 4}

将解分隔开

v_{1} = \frac{- 3 + 5}{2 \cdot 4}, v_{2} = \frac{- 3 - 5}{2 \cdot 4}

v = \frac{- 3 + 5}{2 \cdot 4} : \frac{1}{4}

\frac{- 3 + 5}{2 \cdot 4}

数字相加/相减:

- 3 + 5 = 2

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

数字相乘:

2 \cdot 4 = 8

= \frac{2}{8}

约分:

2

= \frac{1}{4}

v = \frac{- 3 - 5}{2 \cdot 4} : - 1

\frac{- 3 - 5}{2 \cdot 4}

数字相减:

- 3 - 5 = - 8

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 8}{8}

使用分式法则:

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

= - \frac{8}{8}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

v = \frac{1}{4}, v = - 1

v = \frac{1}{4}, v = - 1

代回

v = u^{2}

，求解

u

解

u^{2} = \frac{1}{4} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = \frac{1}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{1}{4}

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

= \frac{1}{2}

使用法则

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

化简

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{1}{4}

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

= \frac{1}{2}

= - \frac{1}{2}

使用法则

1 = 1

= - \frac{1}{2}

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

解

u^{2} = - 1 : u = i, u = - i

u^{2} = - 1

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = - 1, u = - - 1

化简

- 1 : i

- 1

使用虚数运算法则:

- 1 = i

= i

化简

- - 1 : - i

- - 1

使用虚数运算法则:

- 1 = i

= - i

u = i, u = - i

解为

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

u = sin (x)

代回

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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) = i :

无解

sin (x) = i

无解

sin (x) = - i :

无解

sin (x) = - i

无解

合并所有解

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

作图

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