Show Menu
Cheatography
  1. Home >
  2. Education >

Integral Trigonometry Cheat Sheet by

Trigonometric identities and common trigonometric integrals. Note that θ is often interchangeable with x as a variable, excluding trigonometric substitutions. Most important formulas and identities are bolded. Image Sources: 1. https://www.dummies.com/article/academics-the-arts/math/trigonometry/right-triangle-definitions-for-trigonometry-functions-157278/ 2. https://andymath.com/unit-circle/ 3. https://study.com/academy/lesson/graphing-tangent-from-the-unit-circle.html

Basic Trigon­ometric Integrals

∫sin(x)dx
-cos(x)+C
∫cos(x)dx
sin(x)+C
∫sec2(x)dx
tan(x)+C
∫sec(x­)ta­n(x)dx
sec(x)+C
∫csc2(x)dx
-cot(x)+C
∫csc(x­)co­t(x)dx
-csc(x)+C

Common Trigon­ometric Integrals

∫sin(2x)dx
-½cos(­2x)+C
∫cos(2x)dx
½sin(2x)+C = sin(x)­cos­(x)+C
∫tan(x)dx
ln|sec­(x)|+C
∫sec(x)dx
ln|sec­­(x­)­+­ta­­n(x)|+C
∫sec3(x)dx
½(sec(­x)­­tan­­(x­)­+­ln­­|se­­c(­x­)­+t­­an(­x)|)+C
∫csc(x)dx
-ln|cs­­c(­x­)­+c­­ot(­­x)|+C
∫csc3(x)dx
-½(csc­­(­­x­)­­­­­cot­­­­­(­x­­)­­­+­­­ln­­­­­|c­­s­c­­(­­x­­­)­­­+co­­t(­­­x­)|)+C
∫1/(1+x2)dx
arctan­(x)+C
∫1/(a2+x2)dx
(1/a)a­rct­an(­x/a)+C

Secant and Cosecant Integrals

Secant Integral 1
∫sec(x)dx = ln|sec­­­(­x­)­­+­­ta­­­n(­x)|+C
Secant Integral 2
∫sec(x)dx = -ln|se­c(x­)-t­an(­x)|+C
Secant Integral 3
∫sec(x)dx = ½ln|(s­in(­x)+­1)/­(si­n(­­­x)­-1)|+C
Secant Integral 4
∫sec(x)dx = ln|tan­(x/­2+π­/4)|+C
Cosecant Integral 1
∫csc(x)dx = ln|cs­­­c(­­x­­)­-­c­­­ot(­­­x)|+C
Cosecant Integral 2
∫csc(x)dx = -ln|cs­­­c­(­x­­)­­+c­­­ot­(­­­x)|+C
Cosecant Integral 3
∫csc(x)dx = ½ln|(c­os(­x)-­1)/­(c­­­os­(­­­x)+­1)|+C
Cosecant Integral 4
∫csc(x)dx = ln|tan­(x/­2)|+C

Sine and Cosine Unit Circle

  

Powers of Trigon­ometric Functions

Quotient and Reciprocal Identities

Tangent Quotient
tan(x)­=si­n(x­)/c­os(x)
Cotangent Quotient
cot(x)­=co­s(x­)/s­in(x)
 
Sine Reciprocal
sin(x)­=1/­csc(x)
Cosine Reciprocal
cos(x)­=1/­sec(x)
Tangent Reciprocal
tan(x)­=1/­cot(x)
Cosecant Reciprocal
csc(x)­=1/­sin(x)
Secant Reciprocal
sec(x)­=1/­cos(x)
Cotangent Reciprocal
cot(x)­=1/­tan(x)

Pythag­orean Identities

Sine-C­osine Pythag­orean
sin2(x)+cos2(x)=1
Sine Pythag­orean
sin2(x)=1-cos2(x)
Cosine Pythag­orean
cos2(x)=1-sin2(x)
Secant Pythag­orean
tan2(x)+1=sec2(x)
Tangent Pythag­orean
tan2(x)=sec2(x)-1
Secant­-Ta­ngent Pythag­orean
sec2(x)-tan2(x)=1
Cosecant Pythag­orean
1+cot2(x)=csc2(x)
Cotangent Pythag­orean
cot2(x)=csc2(x)-1
Coseca­nt-­Cot­angent Pythag­orean
csc2(x)-cot2(x)=1
The last two triplets of Pythag­orean identities are obtained by dividing all the terms of the original identity by sin²(x) or cos²(x)

Sum and Difference Identities

sin(x+­y)=­sin­(x)­cos(y) + cos(x)­sin(y)
sin(x-­y)=­sin­(x)­cos(y) - cos(x)­sin(y)
cos(x+­y)=­cos­(x)­cos(y) - sin(x)­sin(y)
cos(x-­y)=­cos­(x)­cos(y) + sin(x)­sin(y)

Right-­Tri­angle Trigon­ometric Relations

  

Trigon­ometric Substi­tutions

a2-x2
Let x=asin(θ)
dx=aco­s(θ)dθ
x2-a2
Let x=asec(θ)
dx=ase­c(θ­)ta­n(θ)dθ
x2+a2
Let x=atan(θ)
dx=asec2(θ)dθ
a2-b2x2
Let x=(a/b­)sin(θ)
dx=(a/­b)c­os(θ)dθ
b2x2-a2
Let x=(a/b­)sec(θ)
dx=(a/­b)s­ec(­θ)t­an(θ)dθ
b2x2+a2
Let x=(a/b­)tan(θ)
dx=(a/­b)sec2(θ)dθ
Trigon­ometric substi­tutions are typically used under radicals, however, they are not required to be

For definite integrals, you will need to set x equal to its respective bounds, and solve for θ in order to properly change the bounds of integr­ation with respect to θ

Half-Angle and Double­-Angle Identities

Sine Half-Angle
sin(x/2)=√(½(1-c­os(x)))
Cosine Half-Angle
cos(x/2)=√(½(1+c­os(x)))
Sine Power-­­Re­d­ucing
sin2(x)=½(­1-c­os(2x))
Cosine Power-­­Re­d­ucing
cos2(x)=½(­1+c­os(2x))
 
Sine Double­-Angle
sin(2x­)=2­sin­(x)­cos(x)
Cosine Double­-Angle 1
cos(2x­)=cos2(x)-sin2(x)
Cosine Double­-Angle 2
cos(2x­)=2cos2(x)-1
Cosine Double­-Angle 3
cos(2x­)=1­-2sin2(x)
Sine Power-­­Re­d­ucing and Cosine Power-­­Re­d­ucing identities are variations of the Half-Angle identities

Tangent Unit Circle
                       
 

Metadata

  • Languages:
  • Published: 28th May, 2023
  • Last Updated: 7th October, 2024

Comments

Hey, just stumbled across this. This is AMAZING! Very succinct and has every Trig thing I would need! Just wanted to note, I noticed some overlapping in the Pythagorean Identities Section. It only cuts off stuff on the right of the column. They can be derived from the other stuff, so not a big deal, just noticed! :)

CROSSANT CROSSANT, 13:13 22 Feb 24

Thank you so much!!! I have now edited it to display clearer. I'm glad the cheat sheet was useful!

Add a Comment

Your Comment

Please enter your name.

    Please enter your email address

      Please enter your Comment.

          Related Cheat Sheets

          Integral Cases for Trigonometric Powers Cheat Sheet

          More Cheat Sheets by CROSSANT

          Calculus II Cheat Sheet
          Conic Sections Cheat Sheet
          Trigonometric Properties and Identities Cheat Sheet