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x/sin(x)
Plotting a function graph/ x/sin(x)

Graphing y = x/sin(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
         x   
f(x) = ------
       sin(x)
f(x)=xsin⁡(x)f{\left(x \right)} = \frac{x}{\sin{\left(x \right)}} 
f = x/sin(x)
The graph of the function
02468-8-6-4-2-1010-50005000
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:
xsin⁡(x)=0\frac{x}{\sin{\left(x \right)}} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x/sin(x).
0sin⁡(0)\frac{0}{\sin{\left(0 \right)}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
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
−xcos⁡(x)sin⁡2(x)+1sin⁡(x)=0- \frac{x \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{1}{\sin{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=4.93829501990806⋅10−17x_{1} = 4.93829501990806 \cdot 10^{-17}
x2=−76.9560263103312x_{2} = -76.9560263103312
x3=20.3713029592876x_{3} = 20.3713029592876
x4=39.2444323611642x_{4} = 39.2444323611642
x5=89.5242209304172x_{5} = 89.5242209304172
x6=83.2401924707234x_{6} = 83.2401924707234
x7=−7.72525183693771x_{7} = -7.72525183693771
x8=92.6661922776228x_{8} = 92.6661922776228
x9=36.1006222443756x_{9} = 36.1006222443756
x10=95.8081387868617x_{10} = 95.8081387868617
x11=−17.2207552719308x_{11} = -17.2207552719308
x12=61.2447302603744x_{12} = 61.2447302603744
x13=70.6716857116195x_{13} = 70.6716857116195
x14=10.9041216594289x_{14} = 10.9041216594289
x15=2.01537897347346⋅10−17x_{15} = 2.01537897347346 \cdot 10^{-17}
x16=98.9500628243319x_{16} = 98.9500628243319
x17=80.0981286289451x_{17} = 80.0981286289451
x18=51.8169824872797x_{18} = 51.8169824872797
x19=−10.9041216594289x_{19} = -10.9041216594289
x20=−61.2447302603744x_{20} = -61.2447302603744
x21=−89.5242209304172x_{21} = -89.5242209304172
x22=23.519452498689x_{22} = 23.519452498689
x23=−36.1006222443756x_{23} = -36.1006222443756
x24=−58.1022547544956x_{24} = -58.1022547544956
x25=32.9563890398225x_{25} = 32.9563890398225
x26=29.811598790893x_{26} = 29.811598790893
x27=−83.2401924707234x_{27} = -83.2401924707234
x28=−80.0981286289451x_{28} = -80.0981286289451
x29=−45.5311340139913x_{29} = -45.5311340139913
x30=67.5294347771441x_{30} = 67.5294347771441
x31=−29.811598790893x_{31} = -29.811598790893
x32=17.2207552719308x_{32} = 17.2207552719308
x33=58.1022547544956x_{33} = 58.1022547544956
x34=76.9560263103312x_{34} = 76.9560263103312
x35=−54.9596782878889x_{35} = -54.9596782878889
x36=−67.5294347771441x_{36} = -67.5294347771441
x37=45.5311340139913x_{37} = 45.5311340139913
x38=54.9596782878889x_{38} = 54.9596782878889
x39=7.72525183693771x_{39} = 7.72525183693771
x40=64.3871195905574x_{40} = 64.3871195905574
x41=−20.3713029592876x_{41} = -20.3713029592876
x42=−32.9563890398225x_{42} = -32.9563890398225
x43=−4.49340945790906x_{43} = -4.49340945790906
x44=−95.8081387868617x_{44} = -95.8081387868617
x45=−86.3822220347287x_{45} = -86.3822220347287
x46=−42.3879135681319x_{46} = -42.3879135681319
x47=−98.9500628243319x_{47} = -98.9500628243319
x48=−14.0661939128315x_{48} = -14.0661939128315
x49=−51.8169824872797x_{49} = -51.8169824872797
x50=4.49340945790906x_{50} = 4.49340945790906
x51=86.3822220347287x_{51} = 86.3822220347287
x52=−39.2444323611642x_{52} = -39.2444323611642
x53=−26.6660542588127x_{53} = -26.6660542588127
x54=48.6741442319544x_{54} = 48.6741442319544
x55=−23.519452498689x_{55} = -23.519452498689
x56=−48.6741442319544x_{56} = -48.6741442319544
x57=−64.3871195905574x_{57} = -64.3871195905574
x58=14.0661939128315x_{58} = 14.0661939128315
x59=26.6660542588127x_{59} = 26.6660542588127
x60=73.8138806006806x_{60} = 73.8138806006806
x61=−70.6716857116195x_{61} = -70.6716857116195
x62=42.3879135681319x_{62} = 42.3879135681319
x63=−92.6661922776228x_{63} = -92.6661922776228
x64=−73.8138806006806x_{64} = -73.8138806006806
The values of the extrema at the points:
(4.93829501990806e-17, 1)

(-76.9560263103312, 76.9625232530508)

(20.3713029592876, 20.3958325218432)

(39.2444323611642, 39.2571709544892)

(89.5242209304172, 89.5298058369287)

(83.2401924707234, 83.2461989676591)

(-7.72525183693771, 7.78970576749272)

(92.6661922776228, -92.6715878316184)

(36.1006222443756, -36.1144697653324)

(95.8081387868617, 95.8133574080491)

(-17.2207552719308, -17.2497655675586)

(61.2447302603744, -61.2528936840213)

(70.6716857116195, 70.67876032672)

(10.9041216594289, -10.9498798698263)

(2.01537897347346e-17, 1)

(98.9500628243319, -98.9551157492084)

(80.0981286289451, -80.1043707288125)

(51.8169824872797, 51.8266309351384)

(-10.9041216594289, -10.9498798698263)

(-61.2447302603744, -61.2528936840213)

(-89.5242209304172, 89.5298058369287)

(23.519452498689, -23.5407018977364)

(-36.1006222443756, -36.1144697653324)

(-58.1022547544956, 58.1108596353238)

(32.9563890398225, 32.9715571143392)

(29.811598790893, -29.8283660710601)

(-83.2401924707234, 83.2461989676591)

(-80.0981286289451, -80.1043707288125)

(-45.5311340139913, 45.5421141867616)

(67.5294347771441, -67.5368385499393)

(-29.811598790893, -29.8283660710601)

(17.2207552719308, -17.2497655675586)

(58.1022547544956, 58.1108596353238)

(76.9560263103312, 76.9625232530508)

(-54.9596782878889, -54.9687751137703)

(-67.5294347771441, -67.5368385499393)

(45.5311340139913, 45.5421141867616)

(54.9596782878889, -54.9687751137703)

(7.72525183693771, 7.78970576749272)

(64.3871195905574, 64.3948846506362)

(-20.3713029592876, 20.3958325218432)

(-32.9563890398225, 32.9715571143392)

(-4.49340945790906, -4.6033388487517)

(-95.8081387868617, 95.8133574080491)

(-86.3822220347287, -86.3880100688583)

(-42.3879135681319, -42.399707742618)

(-98.9500628243319, -98.9551157492084)

(-14.0661939128315, 14.1016953304692)

(-51.8169824872797, 51.8266309351384)

(4.49340945790906, -4.6033388487517)

(86.3822220347287, -86.3880100688583)

(-39.2444323611642, 39.2571709544892)

(-26.6660542588127, 26.6847981018021)

(48.6741442319544, -48.6844155424824)

(-23.519452498689, -23.5407018977364)

(-48.6741442319544, -48.6844155424824)

(-64.3871195905574, 64.3948846506362)

(14.0661939128315, 14.1016953304692)

(26.6660542588127, 26.6847981018021)

(73.8138806006806, -73.8206540836068)

(-70.6716857116195, 70.67876032672)

(42.3879135681319, -42.399707742618)

(-92.6661922776228, -92.6715878316184)

(-73.8138806006806, -73.8206540836068)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=4.93829501990806⋅10−17x_{1} = 4.93829501990806 \cdot 10^{-17}
x2=−76.9560263103312x_{2} = -76.9560263103312
x3=20.3713029592876x_{3} = 20.3713029592876
x4=39.2444323611642x_{4} = 39.2444323611642
x5=89.5242209304172x_{5} = 89.5242209304172
x6=83.2401924707234x_{6} = 83.2401924707234
x7=−7.72525183693771x_{7} = -7.72525183693771
x8=95.8081387868617x_{8} = 95.8081387868617
x9=70.6716857116195x_{9} = 70.6716857116195
x10=2.01537897347346⋅10−17x_{10} = 2.01537897347346 \cdot 10^{-17}
x11=51.8169824872797x_{11} = 51.8169824872797
x12=−89.5242209304172x_{12} = -89.5242209304172
x13=−58.1022547544956x_{13} = -58.1022547544956
x14=32.9563890398225x_{14} = 32.9563890398225
x15=−83.2401924707234x_{15} = -83.2401924707234
x16=−45.5311340139913x_{16} = -45.5311340139913
x17=58.1022547544956x_{17} = 58.1022547544956
x18=76.9560263103312x_{18} = 76.9560263103312
x19=45.5311340139913x_{19} = 45.5311340139913
x20=7.72525183693771x_{20} = 7.72525183693771
x21=64.3871195905574x_{21} = 64.3871195905574
x22=−20.3713029592876x_{22} = -20.3713029592876
x23=−32.9563890398225x_{23} = -32.9563890398225
x24=−95.8081387868617x_{24} = -95.8081387868617
x25=−14.0661939128315x_{25} = -14.0661939128315
x26=−51.8169824872797x_{26} = -51.8169824872797
x27=−39.2444323611642x_{27} = -39.2444323611642
x28=−26.6660542588127x_{28} = -26.6660542588127
x29=−64.3871195905574x_{29} = -64.3871195905574
x30=14.0661939128315x_{30} = 14.0661939128315
x31=26.6660542588127x_{31} = 26.6660542588127
x32=−70.6716857116195x_{32} = -70.6716857116195
Maxima of the function at points:
x32=92.6661922776228x_{32} = 92.6661922776228
x32=36.1006222443756x_{32} = 36.1006222443756
x32=−17.2207552719308x_{32} = -17.2207552719308
x32=61.2447302603744x_{32} = 61.2447302603744
x32=10.9041216594289x_{32} = 10.9041216594289
x32=98.9500628243319x_{32} = 98.9500628243319
x32=80.0981286289451x_{32} = 80.0981286289451
x32=−10.9041216594289x_{32} = -10.9041216594289
x32=−61.2447302603744x_{32} = -61.2447302603744
x32=23.519452498689x_{32} = 23.519452498689
x32=−36.1006222443756x_{32} = -36.1006222443756
x32=29.811598790893x_{32} = 29.811598790893
x32=−80.0981286289451x_{32} = -80.0981286289451
x32=67.5294347771441x_{32} = 67.5294347771441
x32=−29.811598790893x_{32} = -29.811598790893
x32=17.2207552719308x_{32} = 17.2207552719308
x32=−54.9596782878889x_{32} = -54.9596782878889
x32=−67.5294347771441x_{32} = -67.5294347771441
x32=54.9596782878889x_{32} = 54.9596782878889
x32=−4.49340945790906x_{32} = -4.49340945790906
x32=−86.3822220347287x_{32} = -86.3822220347287
x32=−42.3879135681319x_{32} = -42.3879135681319
x32=−98.9500628243319x_{32} = -98.9500628243319
x32=4.49340945790906x_{32} = 4.49340945790906
x32=86.3822220347287x_{32} = 86.3822220347287
x32=48.6741442319544x_{32} = 48.6741442319544
x32=−23.519452498689x_{32} = -23.519452498689
x32=−48.6741442319544x_{32} = -48.6741442319544
x32=73.8138806006806x_{32} = 73.8138806006806
x32=42.3879135681319x_{32} = 42.3879135681319
x32=−92.6661922776228x_{32} = -92.6661922776228
x32=−73.8138806006806x_{32} = -73.8138806006806
Decreasing at intervals
[95.8081387868617,∞)\left[95.8081387868617, \infty\right)
Increasing at intervals
(−∞,−95.8081387868617]\left(-\infty, -95.8081387868617\right]
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
x(1+2cos⁡2(x)sin⁡2(x))−2cos⁡(x)sin⁡(x)sin⁡(x)=0\frac{x \left(1 + \frac{2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) - \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)}}}{\sin{\left(x \right)}} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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
lim⁡x→−∞(xsin⁡(x))=lim⁡x→−∞(xsin⁡(x))\lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right) = \lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(xsin⁡(x))y = \lim_{x \to -\infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
lim⁡x→∞(xsin⁡(x))=lim⁡x→∞(xsin⁡(x))\lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right) = \lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(xsin⁡(x))y = \lim_{x \to \infty}\left(\frac{x}{\sin{\left(x \right)}}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞1sin⁡(x)=⟨−∞,∞⟩\lim_{x \to -\infty} \frac{1}{\sin{\left(x \right)}} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞1sin⁡(x)=⟨−∞,∞⟩\lim_{x \to \infty} \frac{1}{\sin{\left(x \right)}} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
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:
xsin⁡(x)=xsin⁡(x)\frac{x}{\sin{\left(x \right)}} = \frac{x}{\sin{\left(x \right)}}
- Yes
xsin⁡(x)=−xsin⁡(x)\frac{x}{\sin{\left(x \right)}} = - \frac{x}{\sin{\left(x \right)}}
- No
so, the function
is
even
The graph
Graphing y = x/sin(x)
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