DuckieDai Exact Trig Values Python Program

trigonometry · MIT License

Overview

Give the exact value of one trig function or all six for a standard angle, find an angle's reference and coterminal angles, and review the special right triangles.

Owner-tested on a TI calculator
Version 1.0.0
2562 bytes
Uses math

Download TRIGEXCT.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 without pi is read as degrees.
Exact values are given for multiples of 30 and 45 degrees, the angles of the special triangles.
Other angles show Not standard angle.

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Learn the math

How to find exact trig values with reference angles

The idea

Every angle can be traced back to an acute angle between it and the x-axis, called the reference angle. The six trig values of an angle are the same as those of its reference angle, except that some signs change, depending on the quadrant the angle ends in.

So to know the exact values of every multiple of 30 and 45 degrees, you only need the special right triangles: the 30-60-90 triangle with sides 1, sqrt(3), and 2, and the 45-45-90 triangle with sides 1, 1, and sqrt(2).

The formula

reference angle: Q1 t, Q2 180 - t, Q3 t - 180, Q4 360 - t

sin 30 = 1/2, sin 45 = sqrt(2)/2, sin 60 = sqrt(3)/2

csc = 1/sin, sec = 1/cos, cot = 1/tan

Q1 to Q4: the four quadrants, numbered counterclockwise from the top right
coterminal: the same angle after whole turns are added or taken away; the program uses the one from 0 up to 360 degrees
undef: a value with 0 on the bottom, such as tan 90

Using the program

Choose 1 for one value: pick the function by number, 1 sin to 6 cot, then type the angle.
Choose 2 for all six values; the program shows sin, cos, and tan, then after Enter csc, sec, and cot.
Type angles in degrees, such as 135, or with pi, such as 5pi/3 or -pi/4. Other angles show Not standard angle.
Choose 3 to see the coterminal angle in degrees and in pi, its quadrant or axis, and its reference angle.
Choose 4 to review the special triangles.

Worked example

Find cos 135. Choose 1, then 2 for cos, and type 135. The angle is in Quadrant 2, its reference angle is 45 degrees, and cosine is negative there.

cos 135 = -cos 45 = -sqrt(2)/2

The program shows cos(135 deg) = and then -sqrt(2)/2.

Find all six values for 5pi/3. Choose 2 and type 5pi/3. The program shows Angle = 300 deg, then sin = -sqrt(3)/2, cos = 1/2, tan = -sqrt(3), and after Enter csc = -2sqrt(3)/3, sec = 2, and cot = -sqrt(3)/3.

Where does -150 degrees end? Choose 3 and type -150. Adding a full turn gives 210, so the program shows Coterminal: 210 = 7pi/6, In: Q3, and Ref angle: 30 = pi/6.

Tips and common mistakes

Find the reference angle first and the sign second. Mixing the two up is the most common mistake.
Remember ASTC for the signs: All in Quadrant 1, Sine in 2, Tangent in 3, and Cosine in 4.
A pi angle needs pi in it: 5pi/6 is 150 degrees, but 5/6 alone is read as 5/6 of a degree.
Angles on an axis, such as 90 or 180, have no quadrant; the program shows axis.

Where this comes up

Reference angles and exact values are taught early in the trigonometry unit of precalculus and algebra 2, and they are expected without a calculator on many tests.

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
N=("sin","cos","tan","csc","sec","cot")
def pg():input("[enter]");print("\n"*11)
def gcd(a,b):
 while b:a,b=b,a%b
 return a
def ex(p,q):
 g=gcd(p,q);p//=g;q//=g;n=p*q;k=1;d=2
 while d*d<=n:
  if n%(d*d):d+=1
  else:n//=d*d;k*=d
 g=gcd(k,q);k//=g;q//=g
 s=str(k) if n==1 else ("" if k==1 else str(k))+"sqrt("+str(n)+")"
 return s if q==1 else s+"/"+str(q)
def ang():
 while 1:
  s=input("Angle (30 or 5pi/6)? ").lower().replace(" ","").replace("*","")
  try:
   if "pi" in s:
    a,b=(s+"/1").split("/")[:2];a=a.replace("pi","")
    v=float(a+"1" if a in("","-") else a)*180/float(b)
   else:v=float(s)
   if v-v:raise ValueError
   return v
  except (ValueError,ZeroDivisionError):print("Like 30 or 5pi/6")
def f(x):return str(int(round(x))) if abs(x-round(x))<1e-7 else str(round(x,4))
def pf(z):
 d=1
 while abs(z*d/180-round(z*d/180))>1e-7 and d<360:d+=1
 n=int(round(z*d/180))
 if d==360:return "?"
 if n==0:return "0"
 return {1:"",-1:"-"}.get(n,str(n))+"pi"+("/"+str(d) if d>1 else "")
def vals(t):
 x=t*pi/180;r=t%180;r=min(r,180-r)
 S=(sin(x)>1e-9)-(sin(x)<-1e-9);C=(cos(x)>1e-9)-(cos(x)<-1e-9)
 if abs(r-round(r))>1e-7:return None
 v={0:(0,1),30:(1,4),45:(1,2),60:(3,4),90:(1,1)}.get(int(round(r)))
 if not v:return None
 p,q=v;c=q-p
 def e(a,b,s):
  if b==0:return "undef"
  if s==0:return "0"
  return ("-" if s<0 else "")+ex(a,b)
 return [e(p,q,S),e(c,q,C),e(p,c,S*C),e(q,p,S),e(q,c,C),e(c,p,S*C)]
def one():
 for i in range(6):print(i+1,N[i])
 n=""
 while len(n)!=1 or n not in "123456":n=input("Function 1-6? ").strip()
 n=int(n)-1;t=ang()%360;V=vals(t);print("\n"*11)
 print(N[n]+"("+f(t)+" deg) =");print(V[n] if V else "Not standard angle");pg()
def six():
 t=ang()%360;V=vals(t);print("\n"*11);print("Angle =",f(t),"deg")
 if not V:print("Not standard angle");pg();return
 for i in range(6):
  print(N[i],"=",V[i])
  if i==2:pg()
 pg()
def ref():
 t=ang()%360;r=t%180;r=min(r,180-r)
 q="axis" if t%90<1e-9 else "Q"+str(int(t//90)+1)
 print("\n"*11);print("Coterminal:",f(t),"=",pf(t));print("In:",q);print("Ref angle:",f(r),"=",pf(r));pg()
def tri():
 print("30-60-90: 1, sqrt(3), 2\n 1 is across from 30\n 2 is the hypotenuse\n45-45-90: 1, 1, sqrt(2)\n sqrt(2) is the hyp");pg()
print("EXACT TRIG VALUES\n      __\n  ___(o )>  quack!\n  \\ <_. )\n   `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
 print("1 One value\n2 All 6 values\n3 Ref + coterminal\n4 Special triangles\n5 Exit")
 c=input("Choice? ").strip();print("\n"*11)
 if c=="5":break
 if c in("1","2","3","4"):(one,six,ref,tri)[int(c)-1]()

from math import sin,cos,pi
N=("sin","cos","tan","csc","sec","cot")
def pg():input("[enter]");print("\n"*11)
def gcd(a,b):
while b:a,b=b,a%b
return a
def ex(p,q):
g=gcd(p,q);p//=g;q//=g;n=p*q;k=1;d=2
while d*d<=n:
if n%(d*d):d+=1
else:n//=d*d;k*=d
g=gcd(k,q);k//=g;q//=g
s=str(k) if n==1 else ("" if k==1 else str(k))+"sqrt("+str(n)+")"
return s if q==1 else s+"/"+str(q)
def ang():
while 1:
s=input("Angle (30 or 5pi/6)? ").lower().replace(" ","").replace("*","")
try:
if "pi" in s:
a,b=(s+"/1").split("/")[:2];a=a.replace("pi","")
v=float(a+"1" if a in("","-") else a)*180/float(b)
else:v=float(s)
if v-v:raise ValueError
return v
except (ValueError,ZeroDivisionError):print("Like 30 or 5pi/6")
def f(x):return str(int(round(x))) if abs(x-round(x))<1e-7 else str(round(x,4))
def pf(z):
d=1
while abs(z*d/180-round(z*d/180))>1e-7 and d<360:d+=1
n=int(round(z*d/180))
if d==360:return "?"
if n==0:return "0"
return {1:"",-1:"-"}.get(n,str(n))+"pi"+("/"+str(d) if d>1 else "")
def vals(t):
x=t*pi/180;r=t%180;r=min(r,180-r)
S=(sin(x)>1e-9)-(sin(x)<-1e-9);C=(cos(x)>1e-9)-(cos(x)<-1e-9)
if abs(r-round(r))>1e-7:return None
v={0:(0,1),30:(1,4),45:(1,2),60:(3,4),90:(1,1)}.get(int(round(r)))
if not v:return None
p,q=v;c=q-p
def e(a,b,s):
if b==0:return "undef"
if s==0:return "0"
return ("-" if s<0 else "")+ex(a,b)
return [e(p,q,S),e(c,q,C),e(p,c,S*C),e(q,p,S),e(q,c,C),e(c,p,S*C)]
def one():
for i in range(6):print(i+1,N[i])
n=""
while len(n)!=1 or n not in "123456":n=input("Function 1-6? ").strip()
n=int(n)-1;t=ang()%360;V=vals(t);print("\n"*11)
print(N[n]+"("+f(t)+" deg) =");print(V[n] if V else "Not standard angle");pg()
def six():
t=ang()%360;V=vals(t);print("\n"*11);print("Angle =",f(t),"deg")
if not V:print("Not standard angle");pg();return
for i in range(6):
print(N[i],"=",V[i])
if i==2:pg()
pg()
def ref():
t=ang()%360;r=t%180;r=min(r,180-r)
q="axis" if t%90<1e-9 else "Q"+str(int(t//90)+1)
print("\n"*11);print("Coterminal:",f(t),"=",pf(t));print("In:",q);print("Ref angle:",f(r),"=",pf(r));pg()
def tri():
print("30-60-90: 1, sqrt(3), 2\n 1 is across from 30\n 2 is the hypotenuse\n45-45-90: 1, 1, sqrt(2)\n sqrt(2) is the hyp");pg()
print("EXACT TRIG VALUES\n __\n ___(o )> quack!\n \\ <_. )\n `---'\nDuckieDai (TM)\nhiddenwordscanner.com");pg()
while 1:
print("1 One value\n2 All 6 values\n3 Ref + coterminal\n4 Special triangles\n5 Exit")
c=input("Choice? ").strip();print("\n"*11)
if c=="5":break
if c in("1","2","3","4"):(one,six,ref,tri)[int(c)-1]()

Filename: TRIGEXCT.py
Size: 2562 bytes
SHA-256: 66d9ce7ffa7933676b83d4782e352a2aaccef6e208faf0e44c8e00f4880925d9

Usage

Inputs

A menu choice: 1 one value, 2 all six values, 3 reference and coterminal angle, 4 special triangles, 5 exit
For one value, the function: 1 sin, 2 cos, 3 tan, 4 csc, 5 sec, 6 cot
An angle in degrees, such as 30, or with pi, such as 5pi/6

Outputs

Exact values such as sqrt(3)/2, -1, or undef
The coterminal angle from 0 up to 360 degrees in degrees and in pi, its quadrant or axis, and its reference angle
The side ratios of the 30-60-90 and 45-45-90 triangles

Units: Angles in degrees or as a multiple of pi.

Assumptions and limitations

Assumptions

An angle without pi is read as degrees.
Exact values are given for multiples of 30 and 45 degrees, the angles of the special triangles.

Constraints

Other angles show Not standard angle.
A pi angle is typed as a fraction such as 5pi/6 or -pi/4; inf and nan are asked for again.

Known failures

A reference angle that is not a whole fraction of pi within 360 parts shows ? for its pi form.
Angles are reduced to 0 up to 360 degrees first, so the program shows 330 deg for -30.

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 before the angle prompt was changed to ask again when inf or nan is typed, where the original stopped with an error. Every other step is unchanged; this exact file has not yet been rerun on a calculator.
Dependencies: math
Suggested calculator name: TRIGEXCT — 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 TRIGEXCT 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 2026-10-22. Download this reviewed version

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