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受欢迎的三角函数 >

4sin^2(x)cot(x)-3cot(x)=0

解答

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

解答

x = \frac{π}{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 = 9 0^{\circ} + 18 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

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

分解

4 sin^{2} (x) cot (x) - 3 cot (x) : cot (x) (2 sin (x) + 3) (2 sin (x) - 3)

4 sin^{2} (x) cot (x) - 3 cot (x)

因式分解出通项

cot (x)

= cot (x) (4 sin^{2} (x) - 3)

分解

4 sin^{2} (x) - 3 : (2 sin (x) + 3) (2 sin (x) - 3)

4 sin^{2} (x) - 3

将

4 sin^{2} (x) - 3

改写为

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

4 sin^{2} (x) - 3

将

4

改写为

2^{2}

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

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用指数法则:

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

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

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

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

使用平方差公式:

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

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

= (2 sin (x) + 3) (2 sin (x) - 3)

= cot (x) (2 sin (x) + 3) (2 sin (x) - 3)

cot (x) (2 sin (x) + 3) (2 sin (x) - 3) = 0

分别求解每个部分

cot (x) = 0 or 2 sin (x) + 3 = 0 or 2 sin (x) - 3 = 0

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = 0

cot (x) = 0

的通解

cot (x)

周期表（周期为

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

2 sin (x) + 3 = 0 : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 sin (x) + 3 = 0

将

3

到右边

2 sin (x) + 3 = 0

两边减去

3

2 sin (x) + 3 - 3 = 0 - 3

化简

2 sin (x) = - 3

2 sin (x) = - 3

两边除以

2

2 sin (x) = - 3

两边除以

2

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

化简

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

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

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

2 sin (x) - 3 = 0

将

3

到右边

2 sin (x) - 3 = 0

两边加上

3

2 sin (x) - 3 + 3 = 0 + 3

化简

2 sin (x) = 3

2 sin (x) = 3

两边除以

2

2 sin (x) = 3

两边除以

2

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

化简

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

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}{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

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

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

cos((2x+3)/2)=-1/2

cos (\frac{2 x + 3}{2}) = - \frac{1}{2}

1=cot^2(x)+csc(x)

1 = cot^{2} (x) + csc (x)

sin(θ/2)=(sqrt(3))/2

sin (\frac{θ}{2}) = \frac{3}{2}

cos (θ) = - 2