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受欢迎的三角函数 >

2cos(x)=cot(x)

解答

2 cos (x) = cot (x)

解答

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

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

2 cos (x) = cot (x)

两边减去

cot (x)

2 cos (x) - cot (x) = 0

用 sin, cos 表示

- cot (x) + 2 cos (x)

使用基本三角恒等式:

cot (x) = \frac{cos ( x )}{sin ( x )}

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

化简

- \frac{cos ( x )}{sin ( x )} + 2 cos (x) : \frac{- cos ( x ) + 2 cos ( x ) sin ( x )}{sin ( x )}

- \frac{cos ( x )}{sin ( x )} + 2 cos (x)

将项转换为分式:

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

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

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

\frac{- cos ( x ) + 2 cos ( x ) sin ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

分解

- cos (x) + 2 cos (x) sin (x) : cos (x) (2 sin (x) - 1)

- cos (x) + 2 cos (x) sin (x)

因式分解出通项

cos (x)

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

cos (x) (2 sin (x) - 1) = 0

分别求解每个部分

cos (x) = 0 or 2 sin (x) - 1 = 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

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

将

1

到右边

2 sin (x) - 1 = 0

两边加上

1

2 sin (x) - 1 + 1 = 0 + 1

化简

2 sin (x) = 1

2 sin (x) = 1

两边除以

2

2 sin (x) = 1

两边除以

2

\frac{2 sin ( x )}{2} = \frac{1}{2}

化简

sin (x) = \frac{1}{2}

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

合并所有解

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

作图

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