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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

Pythag­orean Identities

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

Trigon­ometric Substi­tutions

√(a2-x2) = asin(θ)
√(x2-a2) = asec(θ)
√(x2+a2) = atan(θ)

Right Triangle

  

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)

Common Calculus II 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
∫1/(1+x2)dx
arctan­(x)+C
∫1/(a2+x2)dx
(1/a)a­rct­an(­x/a)+C

Sine and Cosine Unit Circle

  

Sum and Difference

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

Tangent Unit Circle
 

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  • Published: 28th May, 2023

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