Mister Exam

Other calculators

tan(t)
Plotting a function graph/ tan(t)

Graphing y = tan(t)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(t) = tan(t)
f(t)=tan⁡(t)f{\left(t \right)} = \tan{\left(t \right)} 
f = tan(t)
The graph of the function
0102030405060708090-10-200200
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:
tan⁡(t)=0\tan{\left(t \right)} = 0
Solve this equation
The points of intersection with the axis T:

Analytical solution
t1=0t_{1} = 0
Numerical solution
t1=72.2566310325652t_{1} = 72.2566310325652
t2=94.2477796076938t_{2} = 94.2477796076938
t3=−3.14159265358979t_{3} = -3.14159265358979
t4=40.8407044966673t_{4} = 40.8407044966673
t5=−91.106186954104t_{5} = -91.106186954104
t6=−15.707963267949t_{6} = -15.707963267949
t7=−37.6991118430775t_{7} = -37.6991118430775
t8=−31.4159265358979t_{8} = -31.4159265358979
t9=−81.6814089933346t_{9} = -81.6814089933346
t10=6.28318530717959t_{10} = 6.28318530717959
t11=62.8318530717959t_{11} = 62.8318530717959
t12=53.4070751110265t_{12} = 53.4070751110265
t13=81.6814089933346t_{13} = 81.6814089933346
t14=100.530964914873t_{14} = 100.530964914873
t15=50.2654824574367t_{15} = 50.2654824574367
t16=84.8230016469244t_{16} = 84.8230016469244
t17=−47.1238898038469t_{17} = -47.1238898038469
t18=−56.5486677646163t_{18} = -56.5486677646163
t19=−53.4070751110265t_{19} = -53.4070751110265
t20=3.14159265358979t_{20} = 3.14159265358979
t21=25.1327412287183t_{21} = 25.1327412287183
t22=−9.42477796076938t_{22} = -9.42477796076938
t23=−87.9645943005142t_{23} = -87.9645943005142
t24=15.707963267949t_{24} = 15.707963267949
t25=91.106186954104t_{25} = 91.106186954104
t26=97.3893722612836t_{26} = 97.3893722612836
t27=78.5398163397448t_{27} = 78.5398163397448
t28=9.42477796076938t_{28} = 9.42477796076938
t29=−6.28318530717959t_{29} = -6.28318530717959
t30=75.398223686155t_{30} = 75.398223686155
t31=12.5663706143592t_{31} = 12.5663706143592
t32=56.5486677646163t_{32} = 56.5486677646163
t33=−12.5663706143592t_{33} = -12.5663706143592
t34=−34.5575191894877t_{34} = -34.5575191894877
t35=−28.2743338823081t_{35} = -28.2743338823081
t36=18.8495559215388t_{36} = 18.8495559215388
t37=−78.5398163397448t_{37} = -78.5398163397448
t38=−94.2477796076938t_{38} = -94.2477796076938
t39=−25.1327412287183t_{39} = -25.1327412287183
t40=59.6902604182061t_{40} = 59.6902604182061
t41=−21.9911485751286t_{41} = -21.9911485751286
t42=−72.2566310325652t_{42} = -72.2566310325652
t43=−75.398223686155t_{43} = -75.398223686155
t44=31.4159265358979t_{44} = 31.4159265358979
t45=43.9822971502571t_{45} = 43.9822971502571
t46=28.2743338823081t_{46} = 28.2743338823081
t47=−40.8407044966673t_{47} = -40.8407044966673
t48=−50.2654824574367t_{48} = -50.2654824574367
t49=47.1238898038469t_{49} = 47.1238898038469
t50=−84.8230016469244t_{50} = -84.8230016469244
t51=−97.3893722612836t_{51} = -97.3893722612836
t52=−43.9822971502571t_{52} = -43.9822971502571
t53=69.1150383789755t_{53} = 69.1150383789755
t54=−18.8495559215388t_{54} = -18.8495559215388
t55=65.9734457253857t_{55} = 65.9734457253857
t56=−65.9734457253857t_{56} = -65.9734457253857
t57=34.5575191894877t_{57} = 34.5575191894877
t58=0t_{58} = 0
t59=−59.6902604182061t_{59} = -59.6902604182061
t60=−62.8318530717959t_{60} = -62.8318530717959
t61=21.9911485751286t_{61} = 21.9911485751286
t62=87.9645943005142t_{62} = 87.9645943005142
t63=37.6991118430775t_{63} = 37.6991118430775
t64=−100.530964914873t_{64} = -100.530964914873
t65=−69.1150383789755t_{65} = -69.1150383789755
The points of intersection with the Y axis coordinate
The graph crosses Y axis when t equals 0:
substitute t = 0 to tan(t).
tan⁡(0)\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
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
tan⁡2(t)+1=0\tan^{2}{\left(t \right)} + 1 = 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
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(tan⁡2(t)+1)tan⁡(t)=02 \left(\tan^{2}{\left(t \right)} + 1\right) \tan{\left(t \right)} = 0
Solve this equation
The roots of this equation
t1=0t_{1} = 0

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
[0,∞)\left[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at t->+oo and t->-oo
lim⁡t→−∞tan⁡(t)=⟨−∞,∞⟩\lim_{t \to -\infty} \tan{\left(t \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⁡t→∞tan⁡(t)=⟨−∞,∞⟩\lim_{t \to \infty} \tan{\left(t \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(t), divided by t at t->+oo and t ->-oo
lim⁡t→−∞(tan⁡(t)t)=lim⁡t→−∞(tan⁡(t)t)\lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right) = \lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right)
Let's take the limit
so,
inclined asymptote equation on the left:
y=tlim⁡t→−∞(tan⁡(t)t)y = t \lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right)
lim⁡t→∞(tan⁡(t)t)=lim⁡t→∞(tan⁡(t)t)\lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right) = \lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right)
Let's take the limit
so,
inclined asymptote equation on the right:
y=tlim⁡t→∞(tan⁡(t)t)y = t \lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right)
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:
tan⁡(t)=−tan⁡(t)\tan{\left(t \right)} = - \tan{\left(t \right)}
- No
tan⁡(t)=tan⁡(t)\tan{\left(t \right)} = \tan{\left(t \right)}
- Yes
so, the function
is
odd
The graph
Graphing y = tan(t)
    © Mister Exam – Calculator