DuckieDai Exact Trig Expressions Python Program

trigonometry · MIT License

Overview

Work out a whole trig expression, such as 2sin(x)-cos(2x) or sin(75), and show the answer exactly with square roots of 2, 3, and 6 when it has one.

Owner-tested on a TI calculator
Version 1.0.0
3324 bytes
Uses math

Download TRIGEX.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
A number next to a letter or parenthesis is multiplied, so 2x means 2*x and 3sqrt(2) means 3*sqrt(2).
Exact forms use whole numbers from -6 to 6 with sqrt(2), sqrt(3), and sqrt(6), over a bottom up to 12.
Only numbers, pi, x, the listed functions, + - * / ^, and up to nine levels of parentheses are read; anything else shows an error.

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Learn the math

How to work out a trig expression exactly

The idea

Many trig answers are exact numbers made from square roots, such as (sqrt(2)+sqrt(6))/4 for sin 75. A calculator usually shows only a decimal, 0.965926, which hides the exact form you are asked for.

This program works out a whole expression, such as 2sin(x)-cos(2x), and then searches for an exact form made of whole numbers with sqrt(2), sqrt(3), and sqrt(6). Those three roots cover the values of every multiple of 15 degrees, which is where sum and difference identities lead.

The formula

sin(a + b) = sin a cos b + cos a sin b

sin 75 = sin(45 + 30) = (sqrt(2)+sqrt(6))/4

cos(2x) = cos^2 x - sin^2 x

x: a value you type when the expression uses x
^: a power, so 2^3 is 8 and cos(x)^2 is cos squared
~: the decimal value of an exact answer, to six places

Using the program

Choose 1 for radians or 2 for degrees at the start; every angle in that session uses it.
Type an expression with numbers, pi, x, + - * / ^, parentheses, and sin, cos, tan, csc, sec, cot, or sqrt. A number next to a letter or a parenthesis is multiplied, so 2x means 2*x.
If the expression uses x, the program asks x = and you type its value, which can be an expression too.
The answer appears after =, in exact form when one is found, with its decimal after ~.
Press Enter with nothing typed to quit.

Worked example

Find sin 75 exactly. Choose 2 for degrees and type sin(75).

sin(45 + 30) = (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2) = (sqrt(6)+sqrt(2))/4

The program shows = (sqrt(2)+sqrt(6))/4 and ~ 0.965926.

Type 2sin(x)-cos(2x) and then 45 for x. Since cos 90 is 0, the answer is 2 x sqrt(2)/2, and the program shows = sqrt(2) and ~ 1.414214.

Type tan(90). Cosine is 0 at 90 degrees, so the program shows Undefined.

Tips and common mistakes

Check the mode first. sin(30) is 1/2 in degrees but about -0.988032 in radians.
An exact form is found by matching the decimal, so a decimal that happens to land very close to a simple root form is shown in that form. Use the exact answer as a check on your own work.
Use parentheses for function inputs: sin(2x), not sin 2x.
If an expression has a mistake, the program shows Got: with what it read and Error: with what went wrong, then asks again.

Where this comes up

Sum, difference, and double-angle identities are part of the analytic trigonometry unit of precalculus, and exact answers like sin 75 are a common test question.

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,sqrt
k=1
def z(v):return 0 if abs(v)<1e-12 else v
def S(t):return z(sin(t*k))
def C(t):return z(cos(t*k))
F={"sin":S,"cos":C,"tan":lambda t:S(t)/C(t),"csc":lambda t:1/S(t),
"sec":lambda t:1/C(t),"cot":lambda t:C(t)/S(t),"sqrt":sqrt}
G={"pi":pi}
L="abcdefghijklmnopqrstuvwxyz"
D="0123456789."
def fix(s):
 s=s.lower().replace("^","**").replace(" ","")
 t=[];i=0
 while i<len(s):
  j=i+1
  if s[i] in L:
   while j<len(s) and s[j] in L:j+=1
  elif s[i] in D:
   while j<len(s) and s[j] in D:j+=1
  t.append(s[i:j]);i=j
 o=""
 for i in range(len(t)):
  a=t[i-1] if i else "";b=t[i]
  if a and (a[0] in D or a in(")","x","pi")) and (b[0] in L+D or b=="("):o+="*"
  o+=b
 return o
# Reads only numbers, pi, x, + - * / ** and ( ), and the functions in F.
def ev(s):
 t=[];i=0
 while i<len(s):
  j=i+1
  if s[i] in L:
   while j<len(s) and s[j] in L:j+=1
  elif s[i] in D:
   while j<len(s) and s[j] in D:j+=1
  elif s[i:i+2]=="**":j=i+2
  t.append(s[i:j]);i=j
 t.append("");q=[0,0]
 def nx():q[0]+=1;return t[q[0]-1]
 def add():
  v=mul()
  while t[q[0]] in("+","-"):v=v+mul() if nx()=="+" else v-mul()
  return v
 def mul():
  v=un()
  while t[q[0]] in("*","/"):v=v*un() if nx()=="*" else v/un()
  return v
 def un():
  if t[q[0]]=="-":nx();return -un()
  if t[q[0]]=="+":nx();return un()
  v=at()
  if t[q[0]]=="**":
   nx();w=un()
   if v<0:
    if abs(w-round(w))>1e-9:raise ValueError("math domain error")
    w=round(w)
   try:v=v**w
   except OverflowError:raise ValueError("Too large")
  return v
 def par():
  q[1]+=1
  if q[1]>9:raise ValueError("Too many ( )")
  v=add()
  if nx()!=")":raise ValueError("Missing )")
  q[1]-=1;return v
 def at():
  w=nx()
  if w=="(":return par()
  if w[:1] in D:return float(w)
  if w in F:
   if nx()!="(":raise ValueError(w+" needs ( )")
   v=par()
   if v-v or w=="sqrt" and v<0:raise ValueError("math domain error")
   return F[w](v)
  if w in G:return G[w]
  raise ValueError("Unknown: "+(w or "end"))
 v=add()
 if t[q[0]]:raise ValueError("Unexpected: "+t[q[0]])
 return float(v)
def gcd(a,b):
 while b:a,b=b,a%b
 return abs(a)
def tx(w,d):
 g=d
 for n in w:g=gcd(g,n)
 w=[n//g for n in w];d//=g;o=""
 for n,s in zip(w,("","sqrt(2)","sqrt(3)","sqrt(6)")):
  if n:o+=("-" if n<0 else "+" if o else "")+(str(abs(n)) if abs(n)>1 or not s else "")+s
 o=o or "0"
 if d==1:return o
 return ("("+o+")" if "+" in o[1:] or "-" in o[1:] else o)+"/"+str(d)
def nice(v):
 r2=sqrt(2);r3=sqrt(3);r6=sqrt(6)
 for d in range(1,13):
  n=v*d
  for a in range(-6,7):
   for b in range(-6,7):
    p=n-a*r2-b*r3
    if abs(p-round(p))<1e-8:return tx((round(p),a,b,0),d)
  for c in range(-6,7):
   p=n-c*r6
   if abs(p-round(p))<1e-8:return tx((round(p),0,0,c),d)
  for a in range(-6,7):
   for c in range(-6,7):
    p=n-a*r2-c*r6
    if abs(p-round(p))<1e-8:return tx((round(p),a,0,c),d)
print("EXACT TRIG - DuckieDai\nhiddenwordscanner.com\nType like 2sin(x)-cos(2x)\nBlank entry = quit")
if input("1 Radians 2 Degrees? ").strip()=="2":k=pi/180
while 1:
 s=input("Expr? ").strip()
 if not s:break
 e=s
 try:
  e=fix(s)
  if "x" in e:G["x"]=ev(fix(input(" x = ")))
  v=ev(e);r=nice(v)
  print("=",r if r else round(v,6))
  if r and "sqrt" in r:print("~",round(v,6))
 except ZeroDivisionError:print("Undefined")
 except Exception as err:print("Got:",e);print("Error:",err)

from math import sin,cos,pi,sqrt
k=1
def z(v):return 0 if abs(v)<1e-12 else v
def S(t):return z(sin(t*k))
def C(t):return z(cos(t*k))
F={"sin":S,"cos":C,"tan":lambda t:S(t)/C(t),"csc":lambda t:1/S(t),
"sec":lambda t:1/C(t),"cot":lambda t:C(t)/S(t),"sqrt":sqrt}
G={"pi":pi}
L="abcdefghijklmnopqrstuvwxyz"
D="0123456789."
def fix(s):
s=s.lower().replace("^","**").replace(" ","")
t=[];i=0
while i<len(s):
j=i+1
if s[i] in L:
while j<len(s) and s[j] in L:j+=1
elif s[i] in D:
while j<len(s) and s[j] in D:j+=1
t.append(s[i:j]);i=j
o=""
for i in range(len(t)):
a=t[i-1] if i else "";b=t[i]
if a and (a[0] in D or a in(")","x","pi")) and (b[0] in L+D or b=="("):o+="*"
o+=b
return o
# Reads only numbers, pi, x, + - * / ** and ( ), and the functions in F.
def ev(s):
t=[];i=0
while i<len(s):
j=i+1
if s[i] in L:
while j<len(s) and s[j] in L:j+=1
elif s[i] in D:
while j<len(s) and s[j] in D:j+=1
elif s[i:i+2]=="**":j=i+2
t.append(s[i:j]);i=j
t.append("");q=[0,0]
def nx():q[0]+=1;return t[q[0]-1]
def add():
v=mul()
while t[q[0]] in("+","-"):v=v+mul() if nx()=="+" else v-mul()
return v
def mul():
v=un()
while t[q[0]] in("*","/"):v=v*un() if nx()=="*" else v/un()
return v
def un():
if t[q[0]]=="-":nx();return -un()
if t[q[0]]=="+":nx();return un()
v=at()
if t[q[0]]=="**":
nx();w=un()
if v<0:
if abs(w-round(w))>1e-9:raise ValueError("math domain error")
w=round(w)
try:v=v**w
except OverflowError:raise ValueError("Too large")
return v
def par():
q[1]+=1
if q[1]>9:raise ValueError("Too many ( )")
v=add()
if nx()!=")":raise ValueError("Missing )")
q[1]-=1;return v
def at():
w=nx()
if w=="(":return par()
if w[:1] in D:return float(w)
if w in F:
if nx()!="(":raise ValueError(w+" needs ( )")
v=par()
if v-v or w=="sqrt" and v<0:raise ValueError("math domain error")
return F[w](v)
if w in G:return G[w]
raise ValueError("Unknown: "+(w or "end"))
v=add()
if t[q[0]]:raise ValueError("Unexpected: "+t[q[0]])
return float(v)
def gcd(a,b):
while b:a,b=b,a%b
return abs(a)
def tx(w,d):
g=d
for n in w:g=gcd(g,n)
w=[n//g for n in w];d//=g;o=""
for n,s in zip(w,("","sqrt(2)","sqrt(3)","sqrt(6)")):
if n:o+=("-" if n<0 else "+" if o else "")+(str(abs(n)) if abs(n)>1 or not s else "")+s
o=o or "0"
if d==1:return o
return ("("+o+")" if "+" in o[1:] or "-" in o[1:] else o)+"/"+str(d)
def nice(v):
r2=sqrt(2);r3=sqrt(3);r6=sqrt(6)
for d in range(1,13):
n=v*d
for a in range(-6,7):
for b in range(-6,7):
p=n-a*r2-b*r3
if abs(p-round(p))<1e-8:return tx((round(p),a,b,0),d)
for c in range(-6,7):
p=n-c*r6
if abs(p-round(p))<1e-8:return tx((round(p),0,0,c),d)
for a in range(-6,7):
for c in range(-6,7):
p=n-a*r2-c*r6
if abs(p-round(p))<1e-8:return tx((round(p),a,0,c),d)
print("EXACT TRIG - DuckieDai\nhiddenwordscanner.com\nType like 2sin(x)-cos(2x)\nBlank entry = quit")
if input("1 Radians 2 Degrees? ").strip()=="2":k=pi/180
while 1:
s=input("Expr? ").strip()
if not s:break
e=s
try:
e=fix(s)
if "x" in e:G["x"]=ev(fix(input(" x = ")))
v=ev(e);r=nice(v)
print("=",r if r else round(v,6))
if r and "sqrt" in r:print("~",round(v,6))
except ZeroDivisionError:print("Undefined")
except Exception as err:print("Got:",e);print("Error:",err)

Filename: TRIGEX.py
Size: 3324 bytes
SHA-256: b84ebf24999bf37451d730c756475d447bd2be3e7eb751871449bb4e5108a7f5

Usage

Inputs

1 for radians or 2 for degrees
An expression with numbers, pi, x, + - * / ^, parentheses, and sin, cos, tan, csc, sec, cot, and sqrt
A value for x when the expression uses x
Enter with nothing typed to quit

Outputs

The exact answer, such as (sqrt(2)+sqrt(6))/4, followed by its decimal
A decimal to six places when there is no exact form
Undefined for a division by zero, such as tan(90) in degrees

Units: Angles in radians or degrees, chosen at the start.

Assumptions and limitations

Assumptions

A number next to a letter or parenthesis is multiplied, so 2x means 2*x and 3sqrt(2) means 3*sqrt(2).
Exact forms use whole numbers from -6 to 6 with sqrt(2), sqrt(3), and sqrt(6), over a bottom up to 12.

Constraints

Only numbers, pi, x, the listed functions, + - * / ^, and up to nine levels of parentheses are read; anything else shows an error.
A negative number cannot be raised to a fractional power.

Known failures

A decimal that happens to be within 0.00000001 of a simple square-root form is shown in that exact form.
Very large whole numbers are worked out as decimals, so the last digits can differ from the true value.

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 these changes: the website line was added; expressions are read by the program itself instead of eval, so only math can be typed; exact answers also combine sqrt(2) and sqrt(6), so sin(75) shows (sqrt(2)+sqrt(6))/4; and a negative number to a near whole power uses the whole power. Some error messages are worded differently. Every other step is unchanged; this exact file has not yet been rerun on a calculator.
Dependencies: math
Suggested calculator name: TRIGEX — 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 TRIGEX 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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