DuckieDai Trig Identities Python Program

trigonometry · MIT License

Overview

Review 19 basic trig identities, rewrite a function with its cofunction, and check any identity numerically at an angle you choose.

Owner-tested on a TI calculator
Version 1.0.0
3417 bytes
Uses math

Download TRIGID.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, so the cofunction rules use 90-x.
A check passes when the two sides agree to about seven significant digits.
A pi angle is typed as a fraction such as 5pi/6; inf, nan, and angles beyond 1e300 degrees are asked for again.

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Learn the math

How the basic trig identities fit together

The idea

A trig identity is an equation that is true for every angle where both sides are defined. A small set of basic identities does most of the work in simplifying expressions and proving harder identities.

The basic ones come in five families: reciprocal identities, such as csc = 1/sin; quotient identities, such as tan = sin/cos; Pythagorean identities, from sin^2 + cos^2 = 1; even-odd identities, such as cos(-x) = cos x; and cofunction identities, such as sin(90 - x) = cos x.

The formula

sin(x)^2 + cos(x)^2 = 1

1 + tan(x)^2 = sec(x)^2

1 + cot(x)^2 = csc(x)^2

sin(90 - x) = cos(x)

reciprocal: csc, sec, and cot are 1 over sin, cos, and tan
cofunction: sine and cosine, tangent and cotangent, and secant and cosecant swap when the angle is replaced by 90 - x, which is pi/2 - x
MATCH: the two sides agree to about seven significant digits at the angle you chose

Using the program

Choose 1 to page through all 19 identities, eight per screen.
Choose 2 for a cofunction: pick the function, then type the angle. The program rewrites it with the cofunction of the complementary angle, in degrees or in pi.
Choose 3 to check an identity: type its number from the list, then an angle. The program shows the left side, the right side, and MATCH, NO MATCH, or Undefined.
Angles are in degrees unless you type pi, such as pi/6. Press Enter with nothing typed to go back to the menu.

Worked example

Rewrite sin(pi/6) with a cofunction. Choose 2, then 1 for sin, and type pi/6.

sin(pi/6) = cos(pi/2 - pi/6) = cos(pi/3)

The program shows sin(pi/6) = and cos(pi/3). With degrees, tan 20 becomes cot(70).

Check 1 + tan(x)^2 = sec(x)^2 at 60 degrees. Choose 3, type 7, then 60. tan 60 is sqrt(3), so the left side is 1 + 3, and sec 60 is 2, so the right side is 4. The program shows Left = 4, Right = 4, and MATCH.

Check 1/tan(x) = cot(x) at 90 degrees. tan 90 has 0 on the bottom, so the program shows Undefined.

Tips and common mistakes

A MATCH at one angle is evidence, not a proof: a false equation can match at a lucky angle. Try two or three angles, or prove the identity with algebra.
When a check says Undefined, one side divides by zero at that angle. Identities only hold where both sides are defined.
Divide sin^2 + cos^2 = 1 by cos^2 to get 1 + tan^2 = sec^2, or by sin^2 to get 1 + cot^2 = csc^2; you only need to memorise one.
Cofunction angles add up to 90 degrees: sin 20 = cos 70 and tan 35 = cot 55.

Where this comes up

Trig identities fill the analytic trigonometry unit of precalculus and are used again in calculus to simplify integrals and derivatives.

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 as _s,cos as _c,pi
R=("1/sin(x)=csc(x)","1/cos(x)=sec(x)","1/tan(x)=cot(x)",
"sin(x)/cos(x)=tan(x)","cos(x)/sin(x)=cot(x)",
"sin(x)^2+cos(x)^2=1","1+tan(x)^2=sec(x)^2","1+cot(x)^2=csc(x)^2",
"csc(x)^2-cot(x)^2=1","sec(x)^2-tan(x)^2=1",
"sin(-x)=-sin(x)","cos(-x)=cos(x)","tan(-x)=-tan(x)",
"sin(90-x)=cos(x)","cos(90-x)=sin(x)","tan(90-x)=cot(x)",
"sec(90-x)=csc(x)","csc(90-x)=sec(x)","cot(90-x)=tan(x)")
def clr():print("\n"*11)
def pg():input("[enter]");clr()
def ask(p):
 s=input(p).strip()
 if not s:raise KeyError
 return s
def pick(p,n):
 while 1:
  try:
   x=int(ask(p))
   if 1<=x<=n:return x
  except ValueError:pass
  print("Pick 1-"+str(n))
def fmt(x):
 r=round(x)
 return str(int(r)) if abs(x-r)<1e-9 else str(round(x,6))
def ang(p):
 while 1:
  s=ask(p).replace("π","pi").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),1
   else:v=float(s),0
   if v[0]-v[0] or abs(v[0])>1e300:raise ValueError
   return v
  except (ValueError,ZeroDivisionError):print("Bad angle")
def T(f,d):
 v=f(d*pi/180);return 0 if abs(v)<1e-12 else v
S=lambda d:T(_s,d)
C=lambda d:T(_c,d)
G={"sin":S,"cos":C,"tan":lambda d:S(d)/C(d),"csc":lambda d:1/S(d),
"sec":lambda d:1/C(d),"cot":lambda d:C(d)/S(d)}
sin,cos,tan,csc,sec,cot=(G[n] for n in ("sin","cos","tan","csc","sec","cot"))
# Each rule in R as a pair of functions: the left side and the right side.
Q=((lambda x:1/sin(x),lambda x:csc(x)),(lambda x:1/cos(x),lambda x:sec(x)),
(lambda x:1/tan(x),lambda x:cot(x)),(lambda x:sin(x)/cos(x),lambda x:tan(x)),
(lambda x:cos(x)/sin(x),lambda x:cot(x)),(lambda x:sin(x)**2+cos(x)**2,lambda x:1),
(lambda x:1+tan(x)**2,lambda x:sec(x)**2),(lambda x:1+cot(x)**2,lambda x:csc(x)**2),
(lambda x:csc(x)**2-cot(x)**2,lambda x:1),(lambda x:sec(x)**2-tan(x)**2,lambda x:1),
(lambda x:sin(-x),lambda x:-sin(x)),(lambda x:cos(-x),lambda x:cos(x)),
(lambda x:tan(-x),lambda x:-tan(x)),(lambda x:sin(90-x),lambda x:cos(x)),
(lambda x:cos(90-x),lambda x:sin(x)),(lambda x:tan(90-x),lambda x:cot(x)),
(lambda x:sec(90-x),lambda x:csc(x)),(lambda x:csc(90-x),lambda x:sec(x)),
(lambda x:cot(90-x),lambda x:tan(x)))
def show(i):print(i+1,R[i])
def lib():
 for i in range(19):
  show(i)
  if i%8==7:pg()
 pg()
def pf(z):
 d=1
 while abs(z*d/180-round(z*d/180))>1e-9 and d<360:d+=1
 n=round(z*d/180)
 if n==0:return "0"
 return {1:"",-1:"-"}.get(n,str(n))+"pi"+("/"+str(d) if d>1 else "")
def cof():
 N=("sin","cos","tan","csc","sec","cot");O=("cos","sin","cot","sec","csc","tan")
 for i in range(6):print(i+1,N[i])
 f=pick("Function? ",6)-1;x,p=ang("Angle? ");clr()
 z=90-x;h=pf if p else fmt
 print(N[f]+"("+h(x)+") =");print(O[f]+"("+h(z)+")");pg()
def chk():
 k=pick("Rule # (1-19)? ",19)-1;show(k);l,r=Q[k]
 x=ang("Angle? ")[0];clr();show(k)
 try:
  L=l(x);V=r(x)
  print("Left  =",fmt(L));print("Right =",fmt(V))
  print("MATCH" if abs(L-V)<1e-7*max(1,abs(L)) else "NO MATCH")
 except ZeroDivisionError:print("Undefined")
 pg()
def main():
 clr();print("TRIG IDENTITIES\n      __\n  ___(o )>  quack!\n  \\ <_. )\n   `---'")
 print("DuckieDai (TM)\nhiddenwordscanner.com\nBlank entry = back");pg()
 while 1:
  print("TRIG IDENTITIES\n1 All 19 Rules\n2 Cofunction Angle\n3 Numeric Check\n4 Exit")
  try:
   c=pick("Choice? ",4);clr()
   if c==4:break
   (lib,cof,chk)[c-1]()
  except KeyError:clr()
main()

from math import sin as _s,cos as _c,pi
R=("1/sin(x)=csc(x)","1/cos(x)=sec(x)","1/tan(x)=cot(x)",
"sin(x)/cos(x)=tan(x)","cos(x)/sin(x)=cot(x)",
"sin(x)^2+cos(x)^2=1","1+tan(x)^2=sec(x)^2","1+cot(x)^2=csc(x)^2",
"csc(x)^2-cot(x)^2=1","sec(x)^2-tan(x)^2=1",
"sin(-x)=-sin(x)","cos(-x)=cos(x)","tan(-x)=-tan(x)",
"sin(90-x)=cos(x)","cos(90-x)=sin(x)","tan(90-x)=cot(x)",
"sec(90-x)=csc(x)","csc(90-x)=sec(x)","cot(90-x)=tan(x)")
def clr():print("\n"*11)
def pg():input("[enter]");clr()
def ask(p):
s=input(p).strip()
if not s:raise KeyError
return s
def pick(p,n):
while 1:
try:
x=int(ask(p))
if 1<=x<=n:return x
except ValueError:pass
print("Pick 1-"+str(n))
def fmt(x):
r=round(x)
return str(int(r)) if abs(x-r)<1e-9 else str(round(x,6))
def ang(p):
while 1:
s=ask(p).replace("π","pi").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),1
else:v=float(s),0
if v[0]-v[0] or abs(v[0])>1e300:raise ValueError
return v
except (ValueError,ZeroDivisionError):print("Bad angle")
def T(f,d):
v=f(d*pi/180);return 0 if abs(v)<1e-12 else v
S=lambda d:T(_s,d)
C=lambda d:T(_c,d)
G={"sin":S,"cos":C,"tan":lambda d:S(d)/C(d),"csc":lambda d:1/S(d),
"sec":lambda d:1/C(d),"cot":lambda d:C(d)/S(d)}
sin,cos,tan,csc,sec,cot=(G[n] for n in ("sin","cos","tan","csc","sec","cot"))
# Each rule in R as a pair of functions: the left side and the right side.
Q=((lambda x:1/sin(x),lambda x:csc(x)),(lambda x:1/cos(x),lambda x:sec(x)),
(lambda x:1/tan(x),lambda x:cot(x)),(lambda x:sin(x)/cos(x),lambda x:tan(x)),
(lambda x:cos(x)/sin(x),lambda x:cot(x)),(lambda x:sin(x)**2+cos(x)**2,lambda x:1),
(lambda x:1+tan(x)**2,lambda x:sec(x)**2),(lambda x:1+cot(x)**2,lambda x:csc(x)**2),
(lambda x:csc(x)**2-cot(x)**2,lambda x:1),(lambda x:sec(x)**2-tan(x)**2,lambda x:1),
(lambda x:sin(-x),lambda x:-sin(x)),(lambda x:cos(-x),lambda x:cos(x)),
(lambda x:tan(-x),lambda x:-tan(x)),(lambda x:sin(90-x),lambda x:cos(x)),
(lambda x:cos(90-x),lambda x:sin(x)),(lambda x:tan(90-x),lambda x:cot(x)),
(lambda x:sec(90-x),lambda x:csc(x)),(lambda x:csc(90-x),lambda x:sec(x)),
(lambda x:cot(90-x),lambda x:tan(x)))
def show(i):print(i+1,R[i])
def lib():
for i in range(19):
show(i)
if i%8==7:pg()
pg()
def pf(z):
d=1
while abs(z*d/180-round(z*d/180))>1e-9 and d<360:d+=1
n=round(z*d/180)
if n==0:return "0"
return {1:"",-1:"-"}.get(n,str(n))+"pi"+("/"+str(d) if d>1 else "")
def cof():
N=("sin","cos","tan","csc","sec","cot");O=("cos","sin","cot","sec","csc","tan")
for i in range(6):print(i+1,N[i])
f=pick("Function? ",6)-1;x,p=ang("Angle? ");clr()
z=90-x;h=pf if p else fmt
print(N[f]+"("+h(x)+") =");print(O[f]+"("+h(z)+")");pg()
def chk():
k=pick("Rule # (1-19)? ",19)-1;show(k);l,r=Q[k]
x=ang("Angle? ")[0];clr();show(k)
try:
L=l(x);V=r(x)
print("Left =",fmt(L));print("Right =",fmt(V))
print("MATCH" if abs(L-V)<1e-7*max(1,abs(L)) else "NO MATCH")
except ZeroDivisionError:print("Undefined")
pg()
def main():
clr();print("TRIG IDENTITIES\n __\n ___(o )> quack!\n \\ <_. )\n `---'")
print("DuckieDai (TM)\nhiddenwordscanner.com\nBlank entry = back");pg()
while 1:
print("TRIG IDENTITIES\n1 All 19 Rules\n2 Cofunction Angle\n3 Numeric Check\n4 Exit")
try:
c=pick("Choice? ",4);clr()
if c==4:break
(lib,cof,chk)[c-1]()
except KeyError:clr()
main()

Filename: TRIGID.py
Size: 3417 bytes
SHA-256: 0955b601b4f6fa149e2621188ddbc508726f80bc03f9e45906b6486dbd40076c

Usage

Inputs

A menu choice: 1 all 19 rules, 2 cofunction angle, 3 numeric check, 4 exit
For a cofunction, the function: 1 sin to 6 cot, and an angle
For a check, the rule number from 1 to 19 and an angle
Enter with nothing typed goes back to the menu

Outputs

The reciprocal, quotient, Pythagorean, even-odd, and cofunction identities, eight per screen
The cofunction form, such as sin(pi/6) = cos(pi/3)
The left and right side of an identity at your angle, and MATCH, NO MATCH, or Undefined

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

Assumptions and limitations

Assumptions

An angle without pi is read as degrees, so the cofunction rules use 90-x.
A check passes when the two sides agree to about seven significant digits.

Constraints

A pi angle is typed as a fraction such as 5pi/6; inf, nan, and angles beyond 1e300 degrees are asked for again.

Known failures

A numeric check tests one angle, so a MATCH shows the rule holds there, not that it is proved.

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 two changes: the angle prompt now asks again when inf, nan, or an angle beyond 1e300 degrees is typed, where the original stopped with an error, and the numeric check now works each rule out with a fixed pair of functions instead of running its text through Python's eval. The results are the same. Every other step is unchanged; this exact file has not yet been rerun on a calculator.
Dependencies: math
Suggested calculator name: TRIGID — 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 TRIGID 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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