DuckieDai Reference Angles Python Program

trigonometry · MIT License

Overview

Find the quadrant and the reference angle of any angle, in degrees or as a fraction of pi, including the coterminal angle between 0 and 2pi.

Owner-tested on a TI calculator
Version 1.0.0
1894 bytes
No imports

Download REFANGLE.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
An angle in degrees can be any number, such as -150 or 37.5.
A pi angle is typed as a whole-number numerator and denominator, such as 17 and 12 for 17pi/12.
Whole numbers only for a pi numerator and denominator; the denominator cannot be 0.

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Learn the math

How to find the quadrant and reference angle of any angle

The idea

Every angle, however large or negative, ends in one of the four quadrants or on an axis. Adding or taking away full turns does not change where it ends, so the first step is to bring the angle between 0 and 360 degrees, or 0 and 2pi.

The reference angle is the acute angle between where the angle ends and the x-axis. It is the angle you look up in the special triangles; the quadrant then tells you each sign.

The formula

coterminal: add or take away 360 degrees, or 2pi, until 0 <= angle < 360

reference angle: Q I t, Q II 180 - t, Q III t - 180, Q IV 360 - t

in pi: Q I t, Q II pi - t, Q III t - pi, Q IV 2pi - t

Q I to Q IV: the four quadrants, numbered counterclockwise from the top right
coterminal angle: an angle that ends in the same place after whole turns
ON AXIS: an angle that ends on an axis, such as 90 or pi, which is in no quadrant

Using the program

Choose 1 for an angle in degrees and type it, such as -150 or 37.5.
Choose 2 for an angle with pi: type the number in front of pi, then the number under it, so 17pi/12 is 17 and 12.
The program shows the quadrant and the reference angle; for a pi angle it first shows the coterminal angle between 0 and 2pi.
Press Enter to go back to the menu, and choose 3 to exit.

Worked example

Find the reference angle of -150 degrees. Choose 1 and type -150. Adding 360 gives 210, which is in Quadrant 3.

210 - 180 = 30

The program shows Q III REF: 30 deg.

Find the reference angle of 17pi/12. Choose 2 and type 17, then 12. 17pi/12 is between pi and 3pi/2, so it is in Quadrant 3, and 17pi/12 - pi = 5pi/12. The program shows COTERM: 17pi/12 and Q III REF: 5pi/12.

Try -7pi/4: type -7 and 4. Adding 2pi gives pi/4, in Quadrant 1. The program shows COTERM: pi/4 and Q I REF: pi/4.

Tips and common mistakes

The reference angle is never negative and never more than 90 degrees, or pi/2. If yours is, recheck the quadrant.
Measure the reference angle to the x-axis, not the y-axis. In Quadrant 2, 120 degrees has a reference angle of 60, not 30.
For a pi angle, compare the numerator with the denominator: 17pi/12 is more than 12pi/12 = pi and less than 18pi/12 = 3pi/2.
Angles on an axis have no reference angle; the program shows ON AXIS.

Where this comes up

Reference angles open the unit circle unit of precalculus and trigonometry, and every later exact-value and graphing question relies on them.

Exact reviewed bytes

Source code

Download .py

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

# DuckieDai(TM) Reference Angles
# TI-84 Plus CE Python

def gcd(a,b):
 a=abs(a);b=abs(b)
 while b:a,b=b,a%b
 return a

def fmt(x):
 r=round(x,6)
 return str(int(r)) if r==int(r) else str(r)

def inp(p):
 while 1:
  try:return int(input(p))
  except ValueError:print('Whole number please')

def show(n,d):
 if n==0:return '0'
 g=gcd(n,d);n//=g;d//=g
 if d==1:return 'pi' if n==1 else str(n)+'pi'
 return ('pi' if n==1 else str(n)+'pi')+'/'+str(d)

def ref(a,full,half):
 # a is normalized to one revolution
 if a==0 or a==half or a==full//2 or a==full-half:
  return 0,'AXIS'
 if a<half:return a,'I'
 if a<full//2:return full//2-a,'II'
 if a<full-half:return a-full//2,'III'
 return full-a,'IV'

def main():
 print('DuckieDai(TM)')
 print('REFERENCE ANGLES')
 print('   __(o )>')
 print('hiddenwordscanner.com')
 while 1:
  print('1 Degrees  2 Pi fraction')
  print('3 Exit')
  c=input('Choice? ').strip()
  if c=='3':break
  if c=='1':
   while 1:
    try:
     v=float(input('Angle degrees? '))
     if v-v==0:break
    except ValueError:pass
    print('Number please')
   a=v%360
   if a in (0,90,180,270):print('ON AXIS')
   elif a<90:print('Q I   REF:',fmt(a),'deg')
   elif a<180:print('Q II  REF:',fmt(180-a),'deg')
   elif a<270:print('Q III REF:',fmt(a-180),'deg')
   else:print('Q IV  REF:',fmt(360-a),'deg')
  elif c=='2':
   n=inp('Pi numerator? ')
   d=inp('Pi denominator? ')
   if d==0:print('Denominator cannot be 0')
   else:
    if d<0:n=-n;d=-d
    a=n%(2*d)
    print('COTERM:',show(a,d))
    # Avoid floats: work in half-pi units with doubled numerator
    x=2*a
    if x in (0,d,2*d,3*d):print('ON AXIS')
    elif x<d:print('Q I   REF:',show(a,d))
    elif x<2*d:print('Q II  REF:',show(d-a,d))
    elif x<3*d:print('Q III REF:',show(a-d,d))
    else:print('Q IV  REF:',show(2*d-a,d))
  else:print('Choose 1, 2, or 3')
  input('Enter to continue ')
  print('\n'*5)
main()

# DuckieDai(TM) Reference Angles
# TI-84 Plus CE Python
def gcd(a,b):
a=abs(a);b=abs(b)
while b:a,b=b,a%b
return a
def fmt(x):
r=round(x,6)
return str(int(r)) if r==int(r) else str(r)
def inp(p):
while 1:
try:return int(input(p))
except ValueError:print('Whole number please')
def show(n,d):
if n==0:return '0'
g=gcd(n,d);n//=g;d//=g
if d==1:return 'pi' if n==1 else str(n)+'pi'
return ('pi' if n==1 else str(n)+'pi')+'/'+str(d)
def ref(a,full,half):
# a is normalized to one revolution
if a==0 or a==half or a==full//2 or a==full-half:
return 0,'AXIS'
if a<half:return a,'I'
if a<full//2:return full//2-a,'II'
if a<full-half:return a-full//2,'III'
return full-a,'IV'
def main():
print('DuckieDai(TM)')
print('REFERENCE ANGLES')
print(' __(o )>')
print('hiddenwordscanner.com')
while 1:
print('1 Degrees 2 Pi fraction')
print('3 Exit')
c=input('Choice? ').strip()
if c=='3':break
if c=='1':
while 1:
try:
v=float(input('Angle degrees? '))
if v-v==0:break
except ValueError:pass
print('Number please')
a=v%360
if a in (0,90,180,270):print('ON AXIS')
elif a<90:print('Q I REF:',fmt(a),'deg')
elif a<180:print('Q II REF:',fmt(180-a),'deg')
elif a<270:print('Q III REF:',fmt(a-180),'deg')
else:print('Q IV REF:',fmt(360-a),'deg')
elif c=='2':
n=inp('Pi numerator? ')
d=inp('Pi denominator? ')
if d==0:print('Denominator cannot be 0')
else:
if d<0:n=-n;d=-d
a=n%(2*d)
print('COTERM:',show(a,d))
# Avoid floats: work in half-pi units with doubled numerator
x=2*a
if x in (0,d,2*d,3*d):print('ON AXIS')
elif x<d:print('Q I REF:',show(a,d))
elif x<2*d:print('Q II REF:',show(d-a,d))
elif x<3*d:print('Q III REF:',show(a-d,d))
else:print('Q IV REF:',show(2*d-a,d))
else:print('Choose 1, 2, or 3')
input('Enter to continue ')
print('\n'*5)
main()

Filename: REFANGLE.py
Size: 1894 bytes
SHA-256: ba824348a34543bc762fc1f532d781560d3d06fd62b74458ba07a1665bad2d79

Usage

Inputs

A menu choice: 1 degrees, 2 pi fraction, 3 exit
An angle in degrees, or a pi numerator and denominator

Outputs

The quadrant, Q I to Q IV, and the reference angle, or ON AXIS
For a pi angle, the coterminal angle between 0 and 2pi first

Units: Degrees, or multiples of pi.

Assumptions and limitations

Assumptions

An angle in degrees can be any number, such as -150 or 37.5.
A pi angle is typed as a whole-number numerator and denominator, such as 17 and 12 for 17pi/12.

Constraints

Whole numbers only for a pi numerator and denominator; the denominator cannot be 0.
inf and nan are asked for again.

Known failures

Degree answers are rounded to 6 decimal places.
An angle that ends on an axis shows ON AXIS, with no reference angle.

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 the opening screen adds hiddenwordscanner.com, inf and nan are asked for again instead of giving nan, a degree answer such as 30.0 shows as 30, and a menu choice may have spaces around it. This exact file has not yet been rerun on a calculator.
Dependencies: No imports
Suggested calculator name: REFANGLE — 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 REFANGLE 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 2026-10-26. Download this reviewed version

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