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受欢迎的三角函数 >

cos(x)tan^2(x)-3cos(x)=0

解答

cos (x) tan^{2} (x) - 3 cos (x) = 0

解答

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

+1

度数

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

求解步骤

cos (x) tan^{2} (x) - 3 cos (x) = 0

分解

cos (x) tan^{2} (x) - 3 cos (x) : cos (x) (tan (x) + 3) (tan (x) - 3)

cos (x) tan^{2} (x) - 3 cos (x)

因式分解出通项

cos (x)

= cos (x) (tan^{2} (x) - 3)

分解

tan^{2} (x) - 3 : (tan (x) + 3) (tan (x) - 3)

tan^{2} (x) - 3

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用平方差公式:

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

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

= (tan (x) + 3) (tan (x) - 3)

= cos (x) (tan (x) + 3) (tan (x) - 3)

cos (x) (tan (x) + 3) (tan (x) - 3) = 0

分别求解每个部分

cos (x) = 0 or tan (x) + 3 = 0 or tan (x) - 3 = 0

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

cos (x) = 0

cos (x) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

tan (x) + 3 = 0 : x = \frac{2 π}{3} + πn

tan (x) + 3 = 0

将

3

到右边

tan (x) + 3 = 0

两边减去

3

tan (x) + 3 - 3 = 0 - 3

化简

tan (x) = - 3

tan (x) = - 3

tan (x) = - 3

的通解

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

tan (x) - 3 = 0 : x = \frac{π}{3} + πn

tan (x) - 3 = 0

将

3

到右边

tan (x) - 3 = 0

两边加上

3

tan (x) - 3 + 3 = 0 + 3

化简

tan (x) = 3

tan (x) = 3

tan (x) = 3

的通解

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

合并所有解

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

因为方程对以下值无定义:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

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

作图

流行的例子

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

cos^2(θ)-3cos(θ)+2=0

cos(2x-pi/6)=-1/2 , pi/2 <= x<pi

((sin(x)))/((4.2))=((sin(95)))/((10.26))

-sin(x)=-cos(x)