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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

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)

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)

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

Sine and Cosine Unit Circle

 

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)

Powers of Trigon­ometric Functions

Right-­Tri­angle Trigon­ometric Relations

 

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

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

Tangent Unit Circle

                       
 

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!

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