Cheatography

# Integral Trigonometry Cheat Sheet by CROSSANT

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

### 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)

### 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

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! :)

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