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Plotting a function graph/ (cosx)/(sinx)-8

Graphing y = (cosx)/(sinx)-8

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(x)    
f(x) = ------ - 8
       sin(x)
f(x)=−8+cos⁡(x)sin⁡(x)f{\left(x \right)} = -8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} 
f = -8 + cos(x)/sin(x)
The graph of the function
02468-8-6-4-2-1010-10001000
The domain of the function
The points at which the function is not precisely defined:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
−8+cos⁡(x)sin⁡(x)=0-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=atan⁡(18)x_{1} = \operatorname{atan}{\left(\frac{1}{8} \right)}
Numerical solution
x1=−21.8667935805818x_{1} = -21.8667935805818
x2=−43.8579421557103x_{2} = -43.8579421557103
x3=−116.114573188276x_{3} = -116.114573188276
x4=−65.8490907308389x_{4} = -65.8490907308389
x5=−87.8402393059674x_{5} = -87.8402393059674
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)/sin(x) - 8.
−8+cos⁡(0)sin⁡(0)-8 + \frac{\cos{\left(0 \right)}}{\sin{\left(0 \right)}}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
−1−cos⁡2(x)sin⁡2(x)=0-1 - \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(1+cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x)=0\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979

lim⁡x→0−(2(1+cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=−∞\lim_{x \to 0^-}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -\infty
lim⁡x→0+(2(1+cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=∞\lim_{x \to 0^+}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = \infty
- the limits are not equal, so
x1=0x_{1} = 0
- is an inflection point
lim⁡x→3.14159265358979−(2(1+cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=−1.08892367577758⋅1048\lim_{x \to 3.14159265358979^-}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}
lim⁡x→3.14159265358979+(2(1+cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡(x))=−1.08892367577758⋅1048\lim_{x \to 3.14159265358979^+}\left(\frac{2 \left(1 + \frac{\cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = -1.08892367577758 \cdot 10^{48}
- limits are equal, then skip the corresponding point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,−π2]\left(-\infty, - \frac{\pi}{2}\right]
Convex at the intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Vertical asymptotes
Have:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(−8+cos⁡(x)sin⁡(x))y = \lim_{x \to -\infty}\left(-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(−8+cos⁡(x)sin⁡(x))y = \lim_{x \to \infty}\left(-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)/sin(x) - 8, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(−8+cos⁡(x)sin⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(−8+cos⁡(x)sin⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}}{x}\right)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
−8+cos⁡(x)sin⁡(x)=−8−cos⁡(x)sin⁡(x)-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = -8 - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
−8+cos⁡(x)sin⁡(x)=8+cos⁡(x)sin⁡(x)-8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} = 8 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}
- No
so, the function
not is
neither even, nor odd
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