DuckieDai Tan and Cot Graphs Python Program

trigonometry · MIT License

Overview

Graph y = A*tan(Bx - C) + D or y = A*cot(Bx - C) + D: the period, phase shift, midline, asymptotes, and key points, with the parent graph to compare and a drawing on the calculator.

Owner-tested on a TI calculator
Version 1.0.0
2889 bytes
Uses math, ti_plotlib

Download TRIGTC.py Read the source How to install

Before you run it

TI-84 Plus CE Python · No calculator OS or Python App version has been verified.
TI Connect CE .py transfer to the Python App; the owner's transfer details were not recorded
The function is y = A*tan(Bx - C) + D or y = A*cot(Bx - C) + D, with B not 0.
A, B, C, and D may be decimals, fractions such as 1/2, or multiples of pi such as pi/3 or -2pi.
Values are limited to -1,000,000 to 1,000,000; inf and nan are asked for again.

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Learn the math

How to graph tangent and cotangent with transformations

The idea

Tangent and cotangent repeat every pi, not every 2pi like sine and cosine, and they have no maximum or minimum, so there is no amplitude. A stretches the graph up and down instead. Each period sits between two vertical asymptotes, where the graph shoots off to infinity.

In y = A*tan(Bx - C) + D, B squeezes the period, C/B slides the graph left or right, and D moves the midline up or down. Finding five x values per period, two asymptotes and three points, is enough to sketch one full branch.

The formula

period = pi / |B|

phase shift = C / B (right if positive, left if negative)

midline: y = D

tan: asymptotes at the phase shift +/- period/2; cot: asymptotes at the phase shift and phase shift + period

A: the vertical stretch; a negative A flips the graph
B: changes the period to pi/|B|
C: together with B, the phase shift C/B
D: the midline, which the middle point of each branch sits on

Using the program

Choose 1 for tan or 2 for cot.
Type A, B, C, and D. Press Enter to keep the default in brackets. Fractions such as 1/2 and pi values such as pi/2 or -2pi work.
Read the stretch, period, phase shift, and midline, then the two asymptotes and three points of one branch.
The program then shows the same facts for the parent graph y = tan(x) or y = cot(x) so you can compare.
Choose 1 at Draw? to see both graphs on the calculator, yours in blue and the parent in gray, with the key points in red. Press clear to go back to the menu.

Worked example

Graph y = 2tan(2x - pi/2) + 1. Choose 1, then type 2 for A, 2 for B, pi/2 for C, and 1 for D.

period = pi/2, phase shift = (pi/2)/2 = pi/4 right, midline y = 1

The program shows Period: pi/2, Phase: pi/4 right, and Midline: y = 1.

The key x values are a quarter period apart. The program shows VA: x = 0, then (pi/8, -1), (pi/4, 1), (3pi/8, 3), and VA: x = pi/2. Draw the asymptotes as dashed lines, plot the three points, and curve the branch up toward each asymptote.

Tips and common mistakes

Do not look for an amplitude. Tan and cot are stretched by A, but they have no highest or lowest value.
Divide C by B for the shift. In tan(2x - pi/2) the shift is pi/4, not pi/2.
Tan rises from left to right and cot falls, unless A is negative, which flips them.
Asymptotes are where the bottom of the fraction is 0: cos for tan, and sin for cot.

Where this comes up

Graphing tangent and cotangent with period, phase shift, and asymptotes is part of the trigonometric graphs unit of precalculus and trigonometry.

Exact reviewed bytes

Source code

Download .py

View source below, copy it when JavaScript is available, or download the exact .py bytes.

from math import sin,cos,pi
def pg():input('[enter]');print('\n'*11)
def val(t):
 t=t.lower().replace(' ','').replace('*','')
 if 'pi' in t:
  a,b=(t+'/1').split('/')[:2]
  a=a[:-2];a={'':'1','+':'1','-':'-1'}.get(a,a)
  return float(a)*pi/float(b)
 if '/' in t:a,b=t.split('/');return float(a)/float(b)
 return float(t)
def ask(q,d):
 while 1:
  t=input(q+' ['+d+']: ').strip() or d
  try:
   v=val(t)
   if abs(v)<=1e6:return v
  except Exception:pass
  print('Like 2 or pi/3')
def num(x):
 if abs(x-round(x))<1e-8:return str(int(round(x)))
 for d in range(2,33):
  k=round(x*d)
  if abs(x*d-k)<1e-8:return str(int(k))+'/'+str(d)
 return str(round(x,4))
def rad(x):
 if abs(x)<1e-8:return '0'
 z=x/pi
 for d in range(1,65):
  k=int(round(z*d))
  if abs(z*d-k)<1e-8:return {1:'',-1:'-'}.get(k,str(k))+'pi'+('/'+str(d) if d>1 else '')
 return num(x)
N=('tan','cot')
def dn(m,t):return cos(t) if m==0 else sin(t)
def f(m,t):
 d=dn(m,t)
 if abs(d)<1e-8:return None
 return sin(t)/d if m==0 else cos(t)/d
def draw(m,A,B,C,D,P,H,Q):
 try:
  import ti_plotlib as p
  lo=H-(3 if m==0 else 1)*P/4;hi=H+(3 if m==0 else 5)*P/4
  h=max(3,abs(D)+2.6*max(1,abs(A))+1)
  p.cls();p.window(lo,hi,-h,h);p.axes('on')
  try:p.text_at(1,'gray=parent blue=yours','left');p.text_at(12,' '.join(rad(x) for x,y in Q),'left')
  except Exception:pass
  for k in(1,0):
   if k:p.color(140,140,140)
   else:p.color(0,0,255)
   o=None
   for i in range(91):
    x=lo+(hi-lo)*i/90;t=x if k else B*x-C;v=f(m,t)
    y=v if k or v is None else D+A*v
    if y is not None and abs(y)<=h:
     if o and dn(m,o[2])*dn(m,t)>0:p.line(o[0],o[1],x,y,'')
     o=(x,y,t)
    else:o=None
  p.color(255,0,0)
  for x,y in Q:
   if y is not None and -h<=y<=h:p.plot(x,y,'o')
  p.show_plot()
 except Exception as e:print('No plot:',e);pg()
def facts(m,A,B,C,D,T):
 P=pi/abs(B);H=C/B;g=1 if B>0 else -1
 v=([None,-g,0,g,None],[None,g,0,-g,None])[m]
 Q=[(H+(i-(2 if m==0 else 0))*P/4,None if v[i] is None else D+A*v[i]) for i in range(5)]
 print('\n'*11);print(T)
 print('No amplitude\nStretch:',num(abs(A)))
 print('Period:',rad(P))
 print('Phase:',rad(abs(H)),'right' if H>0 else 'left' if H<0 else '')
 print('Midline: y =',num(D))
 if A<0:print('Flipped (A<0)')
 pg();print(T)
 for x,y in Q:print('VA: x = '+rad(x) if y is None else '('+rad(x)+', '+num(y)+')')
 pg();return P,H,Q
print("TAN + COT GRAPHS\n      __\n  ___(o )>  quack!\n  \\ <_. )\n   `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
 m=input('1 tan  2 cot  3 Exit: ').strip()
 if m=='3':break
 if m not in('1','2'):continue
 m=int(m)-1;print('y=A*'+N[m]+'(Bx-C)+D\n[enter] keeps default')
 A=ask('A','1');B=ask('B','1');C=ask('C','0');D=ask('D','0')
 if abs(B)<1e-6:print('B=0?');continue
 P,H,Q=facts(m,A,B,C,D,'YOUR GRAPH (BLUE)')
 facts(m,1,1,0,0,'GRAY: y='+N[m]+'(x)')
 if input('Draw? 1 Yes 2 No: ').strip()=='1':draw(m,A,B,C,D,P,H,Q)
 print('\n'*11)

from math import sin,cos,pi
def pg():input('[enter]');print('\n'*11)
def val(t):
t=t.lower().replace(' ','').replace('*','')
if 'pi' in t:
a,b=(t+'/1').split('/')[:2]
a=a[:-2];a={'':'1','+':'1','-':'-1'}.get(a,a)
return float(a)*pi/float(b)
if '/' in t:a,b=t.split('/');return float(a)/float(b)
return float(t)
def ask(q,d):
while 1:
t=input(q+' ['+d+']: ').strip() or d
try:
v=val(t)
if abs(v)<=1e6:return v
except Exception:pass
print('Like 2 or pi/3')
def num(x):
if abs(x-round(x))<1e-8:return str(int(round(x)))
for d in range(2,33):
k=round(x*d)
if abs(x*d-k)<1e-8:return str(int(k))+'/'+str(d)
return str(round(x,4))
def rad(x):
if abs(x)<1e-8:return '0'
z=x/pi
for d in range(1,65):
k=int(round(z*d))
if abs(z*d-k)<1e-8:return {1:'',-1:'-'}.get(k,str(k))+'pi'+('/'+str(d) if d>1 else '')
return num(x)
N=('tan','cot')
def dn(m,t):return cos(t) if m==0 else sin(t)
def f(m,t):
d=dn(m,t)
if abs(d)<1e-8:return None
return sin(t)/d if m==0 else cos(t)/d
def draw(m,A,B,C,D,P,H,Q):
try:
import ti_plotlib as p
lo=H-(3 if m==0 else 1)*P/4;hi=H+(3 if m==0 else 5)*P/4
h=max(3,abs(D)+2.6*max(1,abs(A))+1)
p.cls();p.window(lo,hi,-h,h);p.axes('on')
try:p.text_at(1,'gray=parent blue=yours','left');p.text_at(12,' '.join(rad(x) for x,y in Q),'left')
except Exception:pass
for k in(1,0):
if k:p.color(140,140,140)
else:p.color(0,0,255)
o=None
for i in range(91):
x=lo+(hi-lo)*i/90;t=x if k else B*x-C;v=f(m,t)
y=v if k or v is None else D+A*v
if y is not None and abs(y)<=h:
if o and dn(m,o[2])*dn(m,t)>0:p.line(o[0],o[1],x,y,'')
o=(x,y,t)
else:o=None
p.color(255,0,0)
for x,y in Q:
if y is not None and -h<=y<=h:p.plot(x,y,'o')
p.show_plot()
except Exception as e:print('No plot:',e);pg()
def facts(m,A,B,C,D,T):
P=pi/abs(B);H=C/B;g=1 if B>0 else -1
v=([None,-g,0,g,None],[None,g,0,-g,None])[m]
Q=[(H+(i-(2 if m==0 else 0))*P/4,None if v[i] is None else D+A*v[i]) for i in range(5)]
print('\n'*11);print(T)
print('No amplitude\nStretch:',num(abs(A)))
print('Period:',rad(P))
print('Phase:',rad(abs(H)),'right' if H>0 else 'left' if H<0 else '')
print('Midline: y =',num(D))
if A<0:print('Flipped (A<0)')
pg();print(T)
for x,y in Q:print('VA: x = '+rad(x) if y is None else '('+rad(x)+', '+num(y)+')')
pg();return P,H,Q
print("TAN + COT GRAPHS\n __\n ___(o )> quack!\n \\ <_. )\n `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
m=input('1 tan 2 cot 3 Exit: ').strip()
if m=='3':break
if m not in('1','2'):continue
m=int(m)-1;print('y=A*'+N[m]+'(Bx-C)+D\n[enter] keeps default')
A=ask('A','1');B=ask('B','1');C=ask('C','0');D=ask('D','0')
if abs(B)<1e-6:print('B=0?');continue
P,H,Q=facts(m,A,B,C,D,'YOUR GRAPH (BLUE)')
facts(m,1,1,0,0,'GRAY: y='+N[m]+'(x)')
if input('Draw? 1 Yes 2 No: ').strip()=='1':draw(m,A,B,C,D,P,H,Q)
print('\n'*11)

Filename: TRIGTC.py
Size: 2889 bytes
SHA-256: 177d9546dc3776d2bee4d7080f1dfce3961e2caabdcb409e1ebc656e1dbaed2a

Usage

Inputs

A choice: 1 tan, 2 cot, 3 exit
A, B, C, and D for y = A*tan(Bx - C) + D; Enter keeps the default shown

Outputs

The stretch, period, phase shift, and midline, and whether the graph is flipped
Five key x values: two vertical asymptotes and three points
The same facts for the parent graph y = tan(x) or y = cot(x)
Optionally, a graph of both on the calculator, with the key points marked

Units: Radians; angles may be typed as multiples of pi.

Assumptions and limitations

Assumptions

The function is y = A*tan(Bx - C) + D or y = A*cot(Bx - C) + D, with B not 0.
A, B, C, and D may be decimals, fractions such as 1/2, or multiples of pi such as pi/3 or -2pi.

Constraints

Values are limited to -1,000,000 to 1,000,000; inf and nan are asked for again.
B must not be 0. A B smaller than 0.000001 is treated as 0.

Known failures

Answers that are not a whole number, a fraction with a denominator up to 32, or a multiple of pi up to pi/64 show 4 decimal places.
The graph needs the calculator's ti_plotlib module; on a computer the program shows No plot instead.

Compatibility and review

Review status: Owner-tested on a TI calculator
Tested on: Owner reports personally running this program on a TI-84 Plus CE Python calculator; the test date, OS version, and Python App version were not recorded
What happened: Owner reported pass on the calculator for the owner's original. For publication, inf, nan, and values beyond 1,000,000 are asked for again instead of stopping the program, and a B smaller than 0.000001 is treated as 0. This exact file has not yet been rerun on a calculator.
Dependencies: math, ti_plotlib
Suggested calculator name: TRIGTC — you can give it another valid, unique name when you transfer it
Desktop test cases: Not available for this interactive-only source

This exact source auto-launches an interactive calculator session, so compatibility evidence comes from the recorded physical-device review rather than an importable desktop fixture.

Transfer and launch

Download the .py file, send it to your calculator with TI Connect CE as TRIGTC or another name, then open the Python App and try the worked example on this page first. This release has no verified calculator OS or Python App version; treat transfer and execution as unverified until independently tested.

Full installation and troubleshooting guide. Check your teacher's or exam's rules before using any program.

Version history

1.0.0 — current version, published 2026-10-31. Download this reviewed version

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