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受欢迎的三角函数 >

5cos(x)+cot(x)=0

解答

5 cos (x) + cot (x) = 0

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.20135 \dots + 2 πn, x = π + 0.20135 \dots + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 11.53695 \dots^{\circ} + 36 0^{\circ} n, x = 191.53695 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 cos (x) + cot (x) = 0

用 sin, cos 表示

cot (x) + 5 cos (x)

使用基本三角恒等式:

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

= \frac{cos ( x )}{sin ( x )} + 5 cos (x)

化简

\frac{cos ( x )}{sin ( x )} + 5 cos (x) : \frac{cos ( x ) + 5 cos ( x ) sin ( x )}{sin ( x )}

\frac{cos ( x )}{sin ( x )} + 5 cos (x)

将项转换为分式:

5 cos (x) = \frac{5 cos ( x ) sin ( x )}{sin ( x )}

= \frac{cos ( x )}{sin ( x )} + \frac{5 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 ) + 5 cos ( x ) sin ( x )}{sin ( x )}

= \frac{cos ( x ) + 5 cos ( x ) sin ( x )}{sin ( x )}

\frac{cos ( x ) + 5 cos ( x ) sin ( x )}{sin ( x )} = 0

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

cos (x) + 5 cos (x) sin (x) = 0

分解

cos (x) + 5 cos (x) sin (x) : cos (x) (5 sin (x) + 1)

cos (x) + 5 cos (x) sin (x)

因式分解出通项

cos (x)

= cos (x) (1 + 5 sin (x))

cos (x) (5 sin (x) + 1) = 0

分别求解每个部分

cos (x) = 0 or 5 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

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

5 sin (x) + 1 = 0

将

1

到右边

5 sin (x) + 1 = 0

两边减去

1

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

化简

5 sin (x) = - 1

5 sin (x) = - 1

两边除以

5

5 sin (x) = - 1

两边除以

5

\frac{5 sin ( x )}{5} = \frac{- 1}{5}

化简

sin (x) = - \frac{1}{5}

sin (x) = - \frac{1}{5}

使用反三角函数性质

sin (x) = - \frac{1}{5}

sin (x) = - \frac{1}{5}

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (- \frac{1}{5}) + 2 πn, x = π + arcsin (\frac{1}{5}) + 2 πn

以小数形式表示解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.20135 \dots + 2 πn, x = π + 0.20135 \dots + 2 πn

作图

流行的例子

tan (θ) = - 0.76

9 tan^{2} (x) = 3

cos^2(x)-10cos(x)-1=0

cos^{2} (x) - 10 cos (x) - 1 = 0

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