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受欢迎的三角函数 >

8sin^3(x)-4sin^2(x)-6sin(x)+3=0

解答

8 sin^{3} (x) - 4 sin^{2} (x) - 6 sin (x) + 3 = 0

解答

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

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

求解步骤

8 sin^{3} (x) - 4 sin^{2} (x) - 6 sin (x) + 3 = 0

用替代法求解

8 sin^{3} (x) - 4 sin^{2} (x) - 6 sin (x) + 3 = 0

令

: sin (x) = u

8 u^{3} - 4 u^{2} - 6 u + 3 = 0

8 u^{3} - 4 u^{2} - 6 u + 3 = 0 : u = \frac{1}{2}, u = - \frac{3}{2}, u = \frac{3}{2}

8 u^{3} - 4 u^{2} - 6 u + 3 = 0

因式分解

8 u^{3} - 4 u^{2} - 6 u + 3 : (2 u - 1) (2 u + 3) (2 u - 3)

8 u^{3} - 4 u^{2} - 6 u + 3

= (8 u^{3} - 4 u^{2}) + (- 6 u + 3)

从

- 6 u + 3

分解出因式

- 3

:

- 3 (2 u - 1)

- 6 u + 3

将

6

改写为

3 \cdot 2

= - 3 \cdot 2 u + 3

因式分解出通项

- 3

= - 3 (2 u - 1)

从

8 u^{3} - 4 u^{2}

分解出因式

4 u^{2}

:

4 u^{2} (2 u - 1)

8 u^{3} - 4 u^{2}

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{3} = u u^{2}

= 8 u u^{2} - 4 u^{2}

将

8

改写为

4 \cdot 2

= 4 \cdot 2 u u^{2} - 4 u^{2}

因式分解出通项

4 u^{2}

= 4 u^{2} (2 u - 1)

= - 3 (2 u - 1) + 4 u^{2} (2 u - 1)

因式分解出通项

2 u - 1

= (2 u - 1) (4 u^{2} - 3)

分解

4 u^{2} - 3 : (2 u + 3) (2 u - 3)

4 u^{2} - 3

将

4 u^{2} - 3

改写为

(2 u)^{2} - (3)^{2}

4 u^{2} - 3

将

4

改写为

2^{2}

= 2^{2} u^{2} - 3

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

2^{2} u^{2} = (2 u)^{2}

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

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

= (2 u + 3) (2 u - 3)

= (2 u - 1) (2 u + 3) (2 u - 3)

(2 u - 1) (2 u + 3) (2 u - 3) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

2 u - 1 = 0 or 2 u + 3 = 0 or 2 u - 3 = 0

解

2 u - 1 = 0 : u = \frac{1}{2}

2 u - 1 = 0

将

1

到右边

2 u - 1 = 0

两边加上

1

2 u - 1 + 1 = 0 + 1

化简

2 u = 1

2 u = 1

两边除以

2

2 u = 1

两边除以

2

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

化简

u = \frac{1}{2}

u = \frac{1}{2}

解

2 u + 3 = 0 : u = - \frac{3}{2}

2 u + 3 = 0

将

3

到右边

2 u + 3 = 0

两边减去

3

2 u + 3 - 3 = 0 - 3

化简

2 u = - 3

2 u = - 3

两边除以

2

2 u = - 3

两边除以

2

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

化简

u = - \frac{3}{2}

u = - \frac{3}{2}

解

2 u - 3 = 0 : u = \frac{3}{2}

2 u - 3 = 0

将

3

到右边

2 u - 3 = 0

两边加上

3

2 u - 3 + 3 = 0 + 3

化简

2 u = 3

2 u = 3

两边除以

2

2 u = 3

两边除以

2

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

化简

u = \frac{3}{2}

u = \frac{3}{2}

解为

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

u = sin (x)

代回

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

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

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{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

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

作图

流行的例子

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2 cos (x) = 1 - sin (x)

tan(2θ)+tan(2θ)=tan(4θ)

tan (2 θ) + tan (2 θ) = tan (4 θ)

1 - 7 cos (θ) = 8

tanh (x) = \frac{12}{13}

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