Mister Exam
Graphing y = sinx/(1-2cosx)
The solution
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sin(x)
f(x) = ------------
1 - 2*cos(x)
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}$$
f = sin(x)/(1 - 2*cos(x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 1.0471975511966$$
$$x_{2} = 5.23598775598299$$
$$x_{1} = 1.0471975511966$$
$$x_{2} = 5.23598775598299$$
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:
$$\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = \pi$$
Numerical solution
$$x_{1} = 72.2566310325652$$
$$x_{2} = -59.6902604182061$$
$$x_{3} = 3.14159265358979$$
$$x_{4} = -43.9822971502571$$
$$x_{5} = 81.6814089933346$$
$$x_{6} = -100.530964914873$$
$$x_{7} = 28.2743338823081$$
$$x_{8} = 65.9734457253857$$
$$x_{9} = -31.4159265358979$$
$$x_{10} = -9.42477796076938$$
$$x_{11} = 40.8407044966673$$
$$x_{12} = 56.5486677646163$$
$$x_{13} = -56.5486677646163$$
$$x_{14} = 12.5663706143592$$
$$x_{15} = 43.9822971502571$$
$$x_{16} = 100.530964914873$$
$$x_{17} = -3.14159265358979$$
$$x_{18} = -15.707963267949$$
$$x_{19} = 59.6902604182061$$
$$x_{20} = 6.28318530717959$$
$$x_{21} = 9.42477796076938$$
$$x_{22} = -53.4070751110265$$
$$x_{23} = -47.1238898038469$$
$$x_{24} = -87.9645943005142$$
$$x_{25} = 69.1150383789755$$
$$x_{26} = 21.9911485751286$$
$$x_{27} = 87.9645943005142$$
$$x_{28} = 18.8495559215388$$
$$x_{29} = -84.8230016469244$$
$$x_{30} = -72.2566310325652$$
$$x_{31} = 25.1327412287183$$
$$x_{32} = 37.6991118430775$$
$$x_{33} = -25.1327412287183$$
$$x_{34} = 0$$
$$x_{35} = 50.2654824574367$$
$$x_{36} = -6.28318530717959$$
$$x_{37} = -65.9734457253857$$
$$x_{38} = -21.9911485751286$$
$$x_{39} = -62.8318530717959$$
$$x_{40} = 75.398223686155$$
$$x_{41} = 84.8230016469244$$
$$x_{42} = 53.4070751110265$$
$$x_{43} = 34.5575191894877$$
$$x_{44} = -28.2743338823081$$
$$x_{45} = 15.707963267949$$
$$x_{46} = -91.106186954104$$
$$x_{47} = 47.1238898038469$$
$$x_{48} = 97.3893722612836$$
$$x_{49} = -69.1150383789755$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = -18.8495559215388$$
$$x_{52} = -50.2654824574367$$
$$x_{53} = -37.6991118430775$$
$$x_{54} = -81.6814089933346$$
$$x_{55} = 62.8318530717959$$
$$x_{56} = 78.5398163397448$$
$$x_{57} = 31.4159265358979$$
$$x_{58} = -78.5398163397448$$
$$x_{59} = -40.8407044966673$$
$$x_{60} = -97.3893722612836$$
$$x_{61} = -75.398223686155$$
$$x_{62} = 91.106186954104$$
$$x_{63} = -12.5663706143592$$
$$x_{64} = -94.2477796076938$$
$$x_{65} = -34.5575191894877$$
so we need to solve the equation:
$$\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = \pi$$
Numerical solution
$$x_{1} = 72.2566310325652$$
$$x_{2} = -59.6902604182061$$
$$x_{3} = 3.14159265358979$$
$$x_{4} = -43.9822971502571$$
$$x_{5} = 81.6814089933346$$
$$x_{6} = -100.530964914873$$
$$x_{7} = 28.2743338823081$$
$$x_{8} = 65.9734457253857$$
$$x_{9} = -31.4159265358979$$
$$x_{10} = -9.42477796076938$$
$$x_{11} = 40.8407044966673$$
$$x_{12} = 56.5486677646163$$
$$x_{13} = -56.5486677646163$$
$$x_{14} = 12.5663706143592$$
$$x_{15} = 43.9822971502571$$
$$x_{16} = 100.530964914873$$
$$x_{17} = -3.14159265358979$$
$$x_{18} = -15.707963267949$$
$$x_{19} = 59.6902604182061$$
$$x_{20} = 6.28318530717959$$
$$x_{21} = 9.42477796076938$$
$$x_{22} = -53.4070751110265$$
$$x_{23} = -47.1238898038469$$
$$x_{24} = -87.9645943005142$$
$$x_{25} = 69.1150383789755$$
$$x_{26} = 21.9911485751286$$
$$x_{27} = 87.9645943005142$$
$$x_{28} = 18.8495559215388$$
$$x_{29} = -84.8230016469244$$
$$x_{30} = -72.2566310325652$$
$$x_{31} = 25.1327412287183$$
$$x_{32} = 37.6991118430775$$
$$x_{33} = -25.1327412287183$$
$$x_{34} = 0$$
$$x_{35} = 50.2654824574367$$
$$x_{36} = -6.28318530717959$$
$$x_{37} = -65.9734457253857$$
$$x_{38} = -21.9911485751286$$
$$x_{39} = -62.8318530717959$$
$$x_{40} = 75.398223686155$$
$$x_{41} = 84.8230016469244$$
$$x_{42} = 53.4070751110265$$
$$x_{43} = 34.5575191894877$$
$$x_{44} = -28.2743338823081$$
$$x_{45} = 15.707963267949$$
$$x_{46} = -91.106186954104$$
$$x_{47} = 47.1238898038469$$
$$x_{48} = 97.3893722612836$$
$$x_{49} = -69.1150383789755$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = -18.8495559215388$$
$$x_{52} = -50.2654824574367$$
$$x_{53} = -37.6991118430775$$
$$x_{54} = -81.6814089933346$$
$$x_{55} = 62.8318530717959$$
$$x_{56} = 78.5398163397448$$
$$x_{57} = 31.4159265358979$$
$$x_{58} = -78.5398163397448$$
$$x_{59} = -40.8407044966673$$
$$x_{60} = -97.3893722612836$$
$$x_{61} = -75.398223686155$$
$$x_{62} = 91.106186954104$$
$$x_{63} = -12.5663706143592$$
$$x_{64} = -94.2477796076938$$
$$x_{65} = -34.5575191894877$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\cos{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} - \frac{2 \sin^{2}{\left(x \right)}}{\left(1 - 2 \cos{\left(x \right)}\right)^{2}} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\cos{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} - \frac{2 \sin^{2}{\left(x \right)}}{\left(1 - 2 \cos{\left(x \right)}\right)^{2}} = 0$$
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
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1.0471975511966$$
$$x_{2} = 5.23598775598299$$
$$\lim_{x \to 1.0471975511966^-}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
$$\lim_{x \to 1.0471975511966^+}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 5.23598775598299^-}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
$$\lim_{x \to 5.23598775598299^+}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
- 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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 1.0471975511966$$
$$x_{2} = 5.23598775598299$$
$$\lim_{x \to 1.0471975511966^-}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
$$\lim_{x \to 1.0471975511966^+}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 5.23598775598299^-}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
$$\lim_{x \to 5.23598775598299^+}\left(\frac{\left(- \frac{2 \left(\cos{\left(x \right)} + \frac{4 \sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right)}{2 \cos{\left(x \right)} - 1} + 1 - \frac{4 \cos{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{2 \cos{\left(x \right)} - 1}\right) = 4.74636579600418 \cdot 10^{47}$$
- 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
$$\left(-\infty, 0\right]$$
Convex at the intervals
$$\left[0, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 1.0471975511966$$
$$x_{2} = 5.23598775598299$$
$$x_{1} = 1.0471975511966$$
$$x_{2} = 5.23598775598299$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)/(1 - 2*cos(x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - 2 \cos{\left(x \right)}\right)}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - 2 \cos{\left(x \right)}\right)}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - 2 \cos{\left(x \right)}\right)}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - 2 \cos{\left(x \right)}\right)}\right)$$
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:
$$\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}$$
- No
$$\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}$$
- No
$$\frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}} = \frac{\sin{\left(x \right)}}{1 - 2 \cos{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd