DuckieDai Sin and Csc Graphs Python Program

trigonometry · MIT License

Overview

Graph y = A*sin(Bx - C) + D or y = A*csc(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
2879 bytes
Uses math, ti_plotlib

Download TRIGSC.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*sin(Bx - C) + D or y = A*csc(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 sine and cosecant with transformations

The idea

Sine waves up and down like cosine, but starts on its midline instead of at its top, so its five key points go midline, top, midline, bottom, midline. Cosecant is 1 divided by sine: U-shaped branches that touch the sine wave at its top and bottom, with asymptotes wherever sine is 0.

In y = A*sin(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. A negative B also turns the wave upside down, because sine of a negative angle is the negative of sine.

The formula

amplitude = |A| (sin only)

period = 2pi / |B|

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

midline: y = D; csc has asymptotes where sin(Bx - C) = 0

A: the amplitude for sin, or the stretch for csc; 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 sin or 2 for csc.
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 csc, three of them are asymptotes.
The program then shows the same facts for the parent graph y = sin(x) or y = csc(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 = 2sin(3x - pi) + 1. Choose 1, then type 2 for A, 3 for B, pi for C, and 1 for D.

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

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

The key points are a quarter period, pi/6, apart, starting on the midline: (pi/3, 1), (pi/2, 3), (2pi/3, 1), (5pi/6, -1), and (pi, 1).

Tips and common mistakes

Sine starts on its midline and goes up first; cosine starts at its top. Mixing them up shifts every point by a quarter period.
Divide C by B for the shift. In sin(3x - pi) the shift is pi/3, not pi.
Cosecant's asymptotes are at the start, middle, and end of each sine period, where the sine wave crosses its midline.
Measure the amplitude from the midline: with D = 1 and A = 2, the top is 3 and the bottom is -1.

Where this comes up

Graphing transformed sine and cosecant waves is the first graph in the trigonometric graphs unit of precalculus and trigonometry, and it returns in physics with waves and sound.

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=('sin','csc')
def f(m,t):
 s=sin(t)
 if m==0:return s
 return None if abs(s)<1e-8 else 1/s
def dn(m,t):return sin(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=([0,g,0,-g,0],[None,g,None,-g,None])[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("SIN + CSC GRAPHS\n      __\n  ___(o )>  quack!\n  \\ <_. )\n   `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
 m=input('1 sin  2 csc  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=('sin','csc')
def f(m,t):
s=sin(t)
if m==0:return s
return None if abs(s)<1e-8 else 1/s
def dn(m,t):return sin(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=([0,g,0,-g,0],[None,g,None,-g,None])[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("SIN + CSC GRAPHS\n __\n ___(o )> quack!\n \\ <_. )\n `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
m=input('1 sin 2 csc 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: TRIGSC.py
Size: 2879 bytes
SHA-256: 8596e4dc24926dcf6d931471a5308a5db4390583c892dc8ea934188689b54ac4

Usage

Inputs

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

Outputs

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