Mister Exam
Other calculators
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- The intersection points of functions
-
How to use it?
- Graphing y =:
-
y^4
-
sin(x)/sqrt(1-cos(x))
- lg(x)
-
sqrt(3)sinx+cosx
- Limit of the function:
-
sin(x)/sqrt(1-cos(x))
-
Identical expressions
- sin(x)/sqrt(one -cos(x))
- sinus of (x) divide by square root of (1 minus co sinus of e of (x))
- sinus of (x) divide by square root of (one minus co sinus of e of (x))
- sin(x)/√(1-cos(x))
- sinx/sqrt1-cosx
- sin(x) divide by sqrt(1-cos(x))
-
Similar expressions
- sin(x)/sqrt(1+cos(x))
- sinx/sqrt(1-cosx)
Graphing y = sin(x)/sqrt(1-cos(x))
The solution
You have entered
[src]
sin(x)
f(x) = --------------
____________
\/ 1 - cos(x)
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}$$
f = sin(x)/(sqrt(1 - cos(x)))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
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:
$$\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = -47.1238898038469$$
$$x_{2} = -72.2566310325652$$
$$x_{3} = -21.9911485751286$$
$$x_{4} = 84.8230016469244$$
$$x_{5} = 53.4070751110265$$
$$x_{6} = -59.6902604182061$$
$$x_{7} = 72.2566310325652$$
$$x_{8} = -9.42477796076938$$
$$x_{9} = -1674.46888436336$$
$$x_{10} = -78.5398163397448$$
$$x_{11} = 78.5398163397448$$
$$x_{12} = 34.5575191894877$$
$$x_{13} = 65.9734457253857$$
$$x_{14} = 91.106186954104$$
$$x_{15} = -3.14159265358979$$
$$x_{16} = -34.5575191894877$$
$$x_{17} = 47.1238898038469$$
$$x_{18} = 9.42477796076938$$
$$x_{19} = -53.4070751110265$$
$$x_{20} = 21.9911485751286$$
$$x_{21} = 40.8407044966673$$
$$x_{22} = 3.14159265358979$$
$$x_{23} = -91.106186954104$$
$$x_{24} = 103.672557568463$$
$$x_{25} = -28.2743338823081$$
$$x_{26} = -9861.45933961836$$
$$x_{27} = 97.3893722612836$$
$$x_{28} = -65.9734457253857$$
$$x_{29} = 28.2743338823081$$
$$x_{30} = 59.6902604182061$$
$$x_{31} = -84.8230016469244$$
$$x_{32} = -122.522113490002$$
$$x_{33} = -15.707963267949$$
$$x_{34} = 15.707963267949$$
$$x_{35} = -97.3893722612836$$
$$x_{36} = -40.8407044966673$$
$$x_{37} = -166.504410640259$$
so we need to solve the equation:
$$\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = -47.1238898038469$$
$$x_{2} = -72.2566310325652$$
$$x_{3} = -21.9911485751286$$
$$x_{4} = 84.8230016469244$$
$$x_{5} = 53.4070751110265$$
$$x_{6} = -59.6902604182061$$
$$x_{7} = 72.2566310325652$$
$$x_{8} = -9.42477796076938$$
$$x_{9} = -1674.46888436336$$
$$x_{10} = -78.5398163397448$$
$$x_{11} = 78.5398163397448$$
$$x_{12} = 34.5575191894877$$
$$x_{13} = 65.9734457253857$$
$$x_{14} = 91.106186954104$$
$$x_{15} = -3.14159265358979$$
$$x_{16} = -34.5575191894877$$
$$x_{17} = 47.1238898038469$$
$$x_{18} = 9.42477796076938$$
$$x_{19} = -53.4070751110265$$
$$x_{20} = 21.9911485751286$$
$$x_{21} = 40.8407044966673$$
$$x_{22} = 3.14159265358979$$
$$x_{23} = -91.106186954104$$
$$x_{24} = 103.672557568463$$
$$x_{25} = -28.2743338823081$$
$$x_{26} = -9861.45933961836$$
$$x_{27} = 97.3893722612836$$
$$x_{28} = -65.9734457253857$$
$$x_{29} = 28.2743338823081$$
$$x_{30} = 59.6902604182061$$
$$x_{31} = -84.8230016469244$$
$$x_{32} = -122.522113490002$$
$$x_{33} = -15.707963267949$$
$$x_{34} = 15.707963267949$$
$$x_{35} = -97.3893722612836$$
$$x_{36} = -40.8407044966673$$
$$x_{37} = -166.504410640259$$
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
$$\frac{\cos{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} - \frac{\sin^{2}{\left(x \right)}}{2 \left(- \cos{\left(x \right)} + 1\right)^{\frac{3}{2}}} = 0$$
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
$$\frac{\cos{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} - \frac{\sin^{2}{\left(x \right)}}{2 \left(- \cos{\left(x \right)} + 1\right)^{\frac{3}{2}}} = 0$$
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
$$- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \pi$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$\lim_{x \to 0^-}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \frac{\sqrt{2}}{4}$$
Let's take the limit
$$\lim_{x \to 0^+}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = - \frac{\sqrt{2}}{4}$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
$$\lim_{x \to 6.28318530717959^-}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = -\infty$$
Let's take the limit
$$\lim_{x \to 6.28318530717959^+}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \infty i$$
Let's take the limit
- the limits are not equal, so
$$x_{2} = 6.28318530717959$$
- is an inflection 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
$$\left[\pi, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \pi\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \pi$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$\lim_{x \to 0^-}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \frac{\sqrt{2}}{4}$$
Let's take the limit
$$\lim_{x \to 0^+}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = - \frac{\sqrt{2}}{4}$$
Let's take the limit
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
$$\lim_{x \to 6.28318530717959^-}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = -\infty$$
Let's take the limit
$$\lim_{x \to 6.28318530717959^+}\left(- \frac{\left(1 + \frac{2 \cos{\left(x \right)} + \frac{3 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \cdot \left(- \cos{\left(x \right)} + 1\right)} + \frac{\cos{\left(x \right)}}{- \cos{\left(x \right)} + 1}\right) \sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \infty i$$
Let's take the limit
- the limits are not equal, so
$$x_{2} = 6.28318530717959$$
- is an inflection 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
$$\left[\pi, \infty\right)$$
Convex at the intervals
$$\left(-\infty, \pi\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(x)/(sqrt(1 - cos(x))), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right) = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right) = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right) = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right) = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \sqrt{- \cos{\left(x \right)} + 1}}\right)$$
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:
$$\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = - \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}$$
- No
$$\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = - \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}$$
- No
$$\frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}} = \frac{\sin{\left(x \right)}}{\sqrt{- \cos{\left(x \right)} + 1}}$$
- No
so, the function
not is
neither even, nor odd
The graph