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

Graphing y = tan(x)*sin(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = tan(x)*sin(x)
f(x)=sin⁡(x)tan⁡(x)f{\left(x \right)} = \sin{\left(x \right)} \tan{\left(x \right)} 
f = sin(x)*tan(x)
The graph of the function
0-60-50-40-30-20-1010203040506070-100100
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:
sin⁡(x)tan⁡(x)=0\sin{\left(x \right)} \tan{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=75.3982240843678x_{1} = 75.3982240843678
x2=−56.5486674066613x_{2} = -56.5486674066613
x3=−65.9734457648377x_{3} = -65.9734457648377
x4=0x_{4} = 0
x5=−6.28318511636354x_{5} = -6.28318511636354
x6=59.6902606242892x_{6} = 59.6902606242892
x7=−100.530964569618x_{7} = -100.530964569618
x8=−15.7079632966714x_{8} = -15.7079632966714
x9=72.2566310277163x_{9} = 72.2566310277163
x10=−3.14159301504925x_{10} = -3.14159301504925
x11=40.8407041479024x_{11} = 40.8407041479024
x12=−50.2654822767396x_{12} = -50.2654822767396
x13=56.5486675928533x_{13} = 56.5486675928533
x14=−75.3982238871507x_{14} = -75.3982238871507
x15=28.274333865158x_{15} = 28.274333865158
x16=3.14159201551055x_{16} = 3.14159201551055
x17=−43.9822971744994x_{17} = -43.9822971744994
x18=−84.8230010248866x_{18} = -84.8230010248866
x19=−97.3893724672266x_{19} = -97.3893724672266
x20=−34.5575188250568x_{20} = -34.5575188250568
x21=9.42477833842133x_{21} = 9.42477833842133
x22=−62.8318524378967x_{22} = -62.8318524378967
x23=81.6814092045399x_{23} = 81.6814092045399
x24=−87.9645943584581x_{24} = -87.9645943584581
x25=−72.2566308569174x_{25} = -72.2566308569174
x26=87.9645943360531x_{26} = 87.9645943360531
x27=−91.1061873420292x_{27} = -91.1061873420292
x28=6.28318528416575x_{28} = 6.28318528416575
x29=62.8318527292552x_{29} = 62.8318527292552
x30=34.5575190128984x_{30} = 34.5575190128984
x31=−59.6902604579627x_{31} = -59.6902604579627
x32=3.14159332506585x_{32} = 3.14159332506585
x33=−94.2477794370922x_{33} = -94.2477794370922
x34=−62.8318537475395x_{34} = -62.8318537475395
x35=−40.8407051595079x_{35} = -40.8407051595079
x36=−28.2743336965558x_{36} = -28.2743336965558
x37=97.3893726665604x_{37} = 97.3893726665604
x38=−69.1150387601409x_{38} = -69.1150387601409
x39=−21.9911485864319x_{39} = -21.9911485864319
x40=84.82300131053x_{40} = 84.82300131053
x41=−84.8230023359423x_{41} = -84.8230023359423
x42=−37.6991118773749x_{42} = -37.6991118773749
x43=91.1061876809771x_{43} = 91.1061876809771
x44=−25.1327415966545x_{44} = -25.1327415966545
x45=50.265482446331x_{45} = 50.265482446331
x46=94.2477796093522x_{46} = 94.2477796093522
x47=−9.42477814706151x_{47} = -9.42477814706151
x48=−18.8495552629365x_{48} = -18.8495552629365
x49=−47.1238901783506x_{49} = -47.1238901783506
x50=21.9911485852154x_{50} = 21.9911485852154
x51=15.7079634638398x_{51} = 15.7079634638398
x52=47.1238891892412x_{52} = 47.1238891892412
x53=69.1150377756597x_{53} = 69.1150377756597
x54=37.6991120440566x_{54} = 37.6991120440566
x55=47.1238905022021x_{55} = 47.1238905022021
x56=53.4070755022832x_{56} = 53.4070755022832
x57=69.1150390913744x_{57} = 69.1150390913744
x58=78.5398161727936x_{58} = 78.5398161727936
x59=25.1327406025294x_{59} = 25.1327406025294
x60=12.5663704329274x_{60} = 12.5663704329274
x61=18.8495555664687x_{61} = 18.8495555664687
x62=43.9822971695024x_{62} = 43.9822971695024
x63=25.1327419134392x_{63} = 25.1327419134392
x64=91.1061863617978x_{64} = 91.1061863617978
x65=−40.8407038505852x_{65} = -40.8407038505852
x66=−18.8495565718301x_{66} = -18.8495565718301
x67=−31.4159267270698x_{67} = -31.4159267270698
x68=31.4159269203024x_{68} = 31.4159269203024
x69=−81.6814090384499x_{69} = -81.6814090384499
x70=−12.5663702433638x_{70} = -12.5663702433638
x71=100.530964752721x_{71} = 100.530964752721
x72=−53.4070753070989x_{72} = -53.4070753070989
x73=−78.5398159881807x_{73} = -78.5398159881807
x74=65.9734457530652x_{74} = 65.9734457530652
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x)*sin(x).
sin⁡(0)tan⁡(0)\sin{\left(0 \right)} \tan{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
(tan⁡2(x)+1)sin⁡(x)+cos⁡(x)tan⁡(x)=0\left(\tan^{2}{\left(x \right)} + 1\right) \sin{\left(x \right)} + \cos{\left(x \right)} \tan{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−πx_{2} = - \pi
x3=πx_{3} = \pi
x4=2πx_{4} = 2 \pi
The values of the extrema at the points:
(0, 0)

(-pi, 0)

(pi, 0)

(2*pi, 0)


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=0x_{1} = 0
x2=2πx_{2} = 2 \pi
Maxima of the function at points:
x2=−πx_{2} = - \pi
x2=πx_{2} = \pi
Decreasing at intervals
[2π,∞)\left[2 \pi, \infty\right)
Increasing at intervals
(−∞,0]∪[π,2π]\left(-\infty, 0\right] \cup \left[\pi, 2 \pi\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(sin⁡(x)tan⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} \tan{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(sin⁡(x)tan⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} \tan{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x)*sin(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(x)tan⁡(x)x)=lim⁡x→−∞(sin⁡(x)tan⁡(x)x)\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \tan{\left(x \right)}}{x}\right) = \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \tan{\left(x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(sin⁡(x)tan⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \tan{\left(x \right)}}{x}\right)
lim⁡x→∞(sin⁡(x)tan⁡(x)x)=lim⁡x→∞(sin⁡(x)tan⁡(x)x)\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \tan{\left(x \right)}}{x}\right) = \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \tan{\left(x \right)}}{x}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(sin⁡(x)tan⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \tan{\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:
sin⁡(x)tan⁡(x)=sin⁡(x)tan⁡(x)\sin{\left(x \right)} \tan{\left(x \right)} = \sin{\left(x \right)} \tan{\left(x \right)}
- No
sin⁡(x)tan⁡(x)=−sin⁡(x)tan⁡(x)\sin{\left(x \right)} \tan{\left(x \right)} = - \sin{\left(x \right)} \tan{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = tan(x)*sin(x)
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