DuckieDai Cos and Sec Graphs Python Program

trigonometry · MIT License

Overview

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

Owner-tested on a TI calculator
Version 1.0.0
2876 bytes
Uses math, ti_plotlib

Download TRIGCS.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*cos(Bx - C) + D or y = A*sec(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 cosine and secant with transformations

The idea

Cosine waves up and down between a highest and lowest value, so it has an amplitude: half the distance from the top to the bottom. Secant is 1 divided by cosine, so it has no amplitude; it has U-shaped branches that touch the cosine wave at its top and bottom and shoot off to infinity wherever cosine is 0.

In y = A*cos(Bx - C) + D, A sets the amplitude, B squeezes the period, C/B slides the graph left or right, and D moves the midline up or down. Five x values a quarter period apart give one full period: the top, the midline, the bottom, the midline, and the top again.

The formula

amplitude = |A| (cos only)

period = 2pi / |B|

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

midline: y = D; sec has asymptotes where cos(Bx - C) = 0

A: the amplitude for cos, or the stretch for sec; a negative A flips the graph
B: changes the period to 2pi/|B|
C: together with B, the phase shift C/B
D: the midline

Using the program

Choose 1 for cos or 2 for sec.
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 amplitude or stretch, period, phase shift, and midline, then five key points across one period. For sec, two of them are the asymptotes.
The program then shows the same facts for the parent graph y = cos(x) or y = sec(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 = 3cos(2x - pi) - 1. Choose 1, then type 3 for A, 2 for B, pi for C, and -1 for D.

amplitude = 3, period = 2pi/2 = pi, phase shift = pi/2 right, midline y = -1

The program shows Amplitude: 3, Period: pi, Phase: pi/2 right, and Midline: y = -1.

The five key points are a quarter period, pi/4, apart: (pi/2, 2), (3pi/4, -1), (pi, -4), (5pi/4, -1), and (3pi/2, 2). The top is 3 above the midline and the bottom 3 below it.

Tips and common mistakes

Measure the amplitude from the midline, not from the x-axis: with D = -1 and A = 3, the top is 2 and the bottom is -4.
Divide C by B for the shift. In cos(2x - pi) the shift is pi/2, not pi.
Secant never crosses its midline. Each branch turns at the top or bottom of the matching cosine wave.
Sketch the cosine wave lightly first; secant's asymptotes are where that wave crosses its midline.

Where this comes up

Graphing transformed cosine and secant waves, with amplitude, period, and phase shift, 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=('cos','sec')
def f(m,t):
 c=cos(t)
 if m==0:return c
 return None if abs(c)<1e-8 else 1/c
def dn(m,t):return cos(t) if m else 1
def draw(m,A,B,C,D,P,H,Q):
 try:
  import ti_plotlib as p
  lo=H-1*P/4;hi=H+5*P/4
  h=max(3,abs(D)+(1 if m==0 else 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=2*pi/abs(B);H=C/B;g=1 if B>0 else -1
 v=([1,0,-1,0,1],[1,None,-1,None,1])[m]
 Q=[(H+(i-0)*P/4,None if v[i] is None else D+A*v[i]) for i in range(5)]
 print('\n'*11);print(T)
 if m==0:print('Amplitude:',num(abs(A)))
 else: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("COS + SEC GRAPHS\n      __\n  ___(o )>  quack!\n  \\ <_. )\n   `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
 m=input('1 cos  2 sec  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=('cos','sec')
def f(m,t):
c=cos(t)
if m==0:return c
return None if abs(c)<1e-8 else 1/c
def dn(m,t):return cos(t) if m else 1
def draw(m,A,B,C,D,P,H,Q):
try:
import ti_plotlib as p
lo=H-1*P/4;hi=H+5*P/4
h=max(3,abs(D)+(1 if m==0 else 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=2*pi/abs(B);H=C/B;g=1 if B>0 else -1
v=([1,0,-1,0,1],[1,None,-1,None,1])[m]
Q=[(H+(i-0)*P/4,None if v[i] is None else D+A*v[i]) for i in range(5)]
print('\n'*11);print(T)
if m==0:print('Amplitude:',num(abs(A)))
else: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("COS + SEC GRAPHS\n __\n ___(o )> quack!\n \\ <_. )\n `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
m=input('1 cos 2 sec 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: TRIGCS.py
Size: 2876 bytes
SHA-256: fc6bdff9d690acedfd53d92d732c3e6f28de32a49b3e695cef785aaa90db1b2f

Usage

Inputs

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

Outputs

The amplitude for cos or the stretch for sec, and the period, phase shift, and midline, and whether the graph is flipped
Five key x values over one period: points, and for sec its two vertical asymptotes
The same facts for the parent graph y = cos(x) or y = sec(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*cos(Bx - C) + D or y = A*sec(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: TRIGCS — 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 TRIGCS 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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