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sint^2
Plotting a function graph/ sint^2

Graphing y = sint^2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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          2   
f(t) = sin (t)
f(t)=sin⁡2(t)f{\left(t \right)} = \sin^{2}{\left(t \right)} 
f = sin(t)^2
The graph of the function
0-60-40-202040608002
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis T at f = 0
so we need to solve the equation:
sin⁡2(t)=0\sin^{2}{\left(t \right)} = 0
Solve this equation
The points of intersection with the axis T:

Analytical solution
t1=0t_{1} = 0
t2=πt_{2} = \pi
Numerical solution
t1=97.3893727097471t_{1} = 97.3893727097471
t2=−12.5663703661411t_{2} = -12.5663703661411
t3=−75.3982238620294t_{3} = -75.3982238620294
t4=−50.2654822953391t_{4} = -50.2654822953391
t5=3.14159244884412t_{5} = 3.14159244884412
t6=−6.28318513794069t_{6} = -6.28318513794069
t7=62.8318528326557t_{7} = 62.8318528326557
t8=78.5398161878405t_{8} = 78.5398161878405
t9=−25.132741473063t_{9} = -25.132741473063
t10=25.1327410188866t_{10} = 25.1327410188866
t11=−59.6902604576401t_{11} = -59.6902604576401
t12=69.1150385885879t_{12} = 69.1150385885879
t13=−100.530964672522t_{13} = -100.530964672522
t14=−34.5575189426108t_{14} = -34.5575189426108
t15=12.5663704518704t_{15} = 12.5663704518704
t16=18.8495554002244t_{16} = 18.8495554002244
t17=−91.106187201329t_{17} = -91.106187201329
t18=−15.7079632965264t_{18} = -15.7079632965264
t19=−84.82300141007t_{19} = -84.82300141007
t20=84.8230014093114t_{20} = 84.8230014093114
t21=28.2743338652012t_{21} = 28.2743338652012
t22=94.2477796093525t_{22} = 94.2477796093525
t23=−31.4159267051849t_{23} = -31.4159267051849
t24=−9.42477812668337t_{24} = -9.42477812668337
t25=−106.814150357553t_{25} = -106.814150357553
t26=56.5486676091327t_{26} = 56.5486676091327
t27=18.8495556796107t_{27} = 18.8495556796107
t28=−31.4159267959754t_{28} = -31.4159267959754
t29=47.123890018392t_{29} = 47.123890018392
t30=53.4070756765307t_{30} = 53.4070756765307
t31=−1734.15914475848t_{31} = -1734.15914475848
t32=69.1150381602162t_{32} = 69.1150381602162
t33=91.1061867314459t_{33} = 91.1061867314459
t34=9.42477821024198t_{34} = 9.42477821024198
t35=31.4159267865366t_{35} = 31.4159267865366
t36=−28.2743337166085t_{36} = -28.2743337166085
t37=−3.14159289677385t_{37} = -3.14159289677385
t38=6.28318528425126t_{38} = 6.28318528425126
t39=−78.5398160958028t_{39} = -78.5398160958028
t40=−72.2566308741333t_{40} = -72.2566308741333
t41=−21.9911485864515t_{41} = -21.9911485864515
t42=−91.1061872003049t_{42} = -91.1061872003049
t43=−18.8495561207399t_{43} = -18.8495561207399
t44=−40.8407042660168t_{44} = -40.8407042660168
t45=87.9645943357576t_{45} = 87.9645943357576
t46=−47.123890151099t_{46} = -47.123890151099
t47=50.2654824463473t_{47} = 50.2654824463473
t48=72.256631027719t_{48} = 72.256631027719
t49=62.8318524523063t_{49} = 62.8318524523063
t50=0t_{50} = 0
t51=84.8230010166547t_{51} = 84.8230010166547
t52=65.9734457528975t_{52} = 65.9734457528975
t53=−87.9645943587732t_{53} = -87.9645943587732
t54=−40.8407046898283t_{54} = -40.8407046898283
t55=−69.1150386737158t_{55} = -69.1150386737158
t56=−84.8230018263493t_{56} = -84.8230018263493
t57=−43.9822971745789t_{57} = -43.9822971745789
t58=−34.5575189701076t_{58} = -34.5575189701076
t59=−69.1150386253436t_{59} = -69.1150386253436
t60=−62.8318528379059t_{60} = -62.8318528379059
t61=75.3982241944528t_{61} = 75.3982241944528
t62=−97.3893724403711t_{62} = -97.3893724403711
t63=−3.14159311568248t_{63} = -3.14159311568248
t64=53.4070753627408t_{64} = 53.4070753627408
t65=40.8407042560881t_{65} = 40.8407042560881
t66=25.1327414478072t_{66} = 25.1327414478072
t67=−94.2477794529919t_{67} = -94.2477794529919
t68=37.6991120192083t_{68} = 37.6991120192083
t69=97.3893725148693t_{69} = 97.3893725148693
t70=81.6814091761104t_{70} = 81.6814091761104
t71=43.982297169427t_{71} = 43.982297169427
t72=−12.5663700417108t_{72} = -12.5663700417108
t73=−53.4070752836338t_{73} = -53.4070752836338
t74=31.4159271479423t_{74} = 31.4159271479423
t75=59.6902605976901t_{75} = 59.6902605976901
t76=91.1061871583643t_{76} = 91.1061871583643
t77=34.5575190304759t_{77} = 34.5575190304759
t78=15.7079634406648t_{78} = 15.7079634406648
t79=−37.6991118771514t_{79} = -37.6991118771514
t80=−81.6814090380061t_{80} = -81.6814090380061
t81=9.42477859080277t_{81} = 9.42477859080277
t82=−25.132741632083t_{82} = -25.132741632083
t83=21.9911485851964t_{83} = 21.9911485851964
t84=40.840703919946t_{84} = 40.840703919946
t85=−65.9734457650176t_{85} = -65.9734457650176
t86=−18.8495556944209t_{86} = -18.8495556944209
t87=−47.1238900492539t_{87} = -47.1238900492539
t88=47.123889589354t_{88} = 47.123889589354
t89=3.14159287686128t_{89} = 3.14159287686128
t90=−62.8318532583801t_{90} = -62.8318532583801
t91=−56.5486675191652t_{91} = -56.5486675191652
t92=75.3982239388525t_{92} = 75.3982239388525
t93=100.530964766599t_{93} = 100.530964766599
The points of intersection with the Y axis coordinate
The graph crosses Y axis when t equals 0:
substitute t = 0 to sin(t)^2.
sin⁡2(0)\sin^{2}{\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
ddtf(t)=0\frac{d}{d t} f{\left(t \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddtf(t)=\frac{d}{d t} f{\left(t \right)} =
the first derivative
2sin⁡(t)cos⁡(t)=02 \sin{\left(t \right)} \cos{\left(t \right)} = 0
Solve this equation
The roots of this equation
t1=0t_{1} = 0
t2=π2t_{2} = \frac{\pi}{2}
t3=πt_{3} = \pi
t4=3π2t_{4} = \frac{3 \pi}{2}
The values of the extrema at the points:
(0, 0)

 pi    
(--, 1)
 2     

(pi, 0)

 3*pi    
(----, 1)
  2


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:
t1=0t_{1} = 0
t2=πt_{2} = \pi
Maxima of the function at points:
t2=π2t_{2} = \frac{\pi}{2}
t2=3π2t_{2} = \frac{3 \pi}{2}
Decreasing at intervals
[0,π2]∪[π,∞)\left[0, \frac{\pi}{2}\right] \cup \left[\pi, \infty\right)
Increasing at intervals
(−∞,0]\left(-\infty, 0\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dt2f(t)=0\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dt2f(t)=\frac{d^{2}}{d t^{2}} f{\left(t \right)} =
the second derivative
2(−sin⁡2(t)+cos⁡2(t))=02 \left(- \sin^{2}{\left(t \right)} + \cos^{2}{\left(t \right)}\right) = 0
Solve this equation
The roots of this equation
t1=−π4t_{1} = - \frac{\pi}{4}
t2=π4t_{2} = \frac{\pi}{4}

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
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]
Convex at the intervals
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at t->+oo and t->-oo
lim⁡t→−∞sin⁡2(t)=⟨0,1⟩\lim_{t \to -\infty} \sin^{2}{\left(t \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
lim⁡t→∞sin⁡2(t)=⟨0,1⟩\lim_{t \to \infty} \sin^{2}{\left(t \right)} = \left\langle 0, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨0,1⟩y = \left\langle 0, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(t)^2, divided by t at t->+oo and t ->-oo
lim⁡t→−∞(sin⁡2(t)t)=0\lim_{t \to -\infty}\left(\frac{\sin^{2}{\left(t \right)}}{t}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡t→∞(sin⁡2(t)t)=0\lim_{t \to \infty}\left(\frac{\sin^{2}{\left(t \right)}}{t}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-t) и f = -f(-t).
So, check:
sin⁡2(t)=sin⁡2(t)\sin^{2}{\left(t \right)} = \sin^{2}{\left(t \right)}
- Yes
sin⁡2(t)=−sin⁡2(t)\sin^{2}{\left(t \right)} = - \sin^{2}{\left(t \right)}
- No
so, the function
is
even
The graph
Graphing y = sint^2
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