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受欢迎的三角函数 >

5cos(x)tan(x)+3tan(x)=0

解答

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

解答

x = πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = - 126.86989 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

分解

5 cos (x) tan (x) + 3 tan (x) : tan (x) (5 cos (x) + 3)

5 cos (x) tan (x) + 3 tan (x)

因式分解出通项

tan (x)

= tan (x) (5 cos (x) + 3)

tan (x) (5 cos (x) + 3) = 0

分别求解每个部分

tan (x) = 0 or 5 cos (x) + 3 = 0

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 0

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 = 0 + πn

x = 0 + πn

解

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

5 cos (x) + 3 = 0 : x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

5 cos (x) + 3 = 0

将

3

到右边

5 cos (x) + 3 = 0

两边减去

3

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

化简

5 cos (x) = - 3

5 cos (x) = - 3

两边除以

5

5 cos (x) = - 3

两边除以

5

\frac{5 cos ( x )}{5} = \frac{- 3}{5}

化简

cos (x) = - \frac{3}{5}

cos (x) = - \frac{3}{5}

使用反三角函数性质

cos (x) = - \frac{3}{5}

cos (x) = - \frac{3}{5}

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

合并所有解

x = πn, x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

以小数形式表示解

x = πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

作图

流行的例子

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4 cos (x) tan (x) = 3 tan (x)

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2sin(x-pi/6)+1=0

2 sin (x - \frac{π}{6}) + 1 = 0

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3 - 3 sin (θ) = 6 cos^{2} (θ)