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
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∫sin(x)dx
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-cos(x)+C
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∫cos(x)dx
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sin(x)+C
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∫sec2(x)dx
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tan(x)+C
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∫sec(x)tan(x)dx
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sec(x)+C
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∫csc2(x)dx
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-cot(x)+C
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∫csc(x)cot(x)dx
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-csc(x)+C
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∫sin(2x)dx
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-½cos(2x)+C
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∫cos(2x)dx
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½sin(2x)+C = sin(x)cos(x)+C
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∫tan(x)dx
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ln|sec(x)|+C
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∫sec(x)dx
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ln|sec(x)+tan(x)|+C
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∫sec3(x)dx
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½(sec(x)tan(x)+ln|sec(x)+tan(x)|)+C
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∫csc(x)dx
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-ln|csc(x)+cot(x)|+C
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∫csc3(x)dx
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-½(csc(x)cot(x)+ln|csc(x)+cot(x)|)+C
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∫1/(1+x2)dx
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arctan(x)+C
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∫1/(a2+x2)dx
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(1/a)arctan(x/a)+C
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Secant Integral 1
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∫sec(x)dx = ln|sec(x)+tan(x)|+C
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Secant Integral 2
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∫sec(x)dx = -ln|sec(x)-tan(x)|+C
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Secant Integral 3
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∫sec(x)dx = ½ln|(sin(x)+1)/(sin(x)-1)|+C
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Secant Integral 4
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∫sec(x)dx = ln|tan(x/2+π/4)|+C
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Cosecant Integral 1
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∫csc(x)dx = -ln|csc(x)+cot(x)|+C
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Cosecant Integral 2
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∫csc(x)dx = ln|csc(x)-cot(x)|+C
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Cosecant Integral 3
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∫csc(x)dx = ½ln|(cos(x)-1)/(cos(x)+1)|+C
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Cosecant Integral 4
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∫csc(x)dx = ln|tan(x/2)|+C
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Tangent Quotient
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tan(x)=sin(x)/cos(x)
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Cotangent Quotient
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cot(x)=cos(x)/sin(x)
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Sine Reciprocal
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sin(x)=1/csc(x)
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Cosine Reciprocal
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cos(x)=1/sec(x)
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Tangent Reciprocal
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tan(x)=1/cot(x)
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Cosecant Reciprocal
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csc(x)=1/sin(x)
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Secant Reciprocal
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sec(x)=1/cos(x)
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Cotangent Reciprocal
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cot(x)=1/tan(x)
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Sine-Cosine Pythagorean
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sin2(x)+cos2(x)=1
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Sine Pythagorean
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sin2(x)=1-cos2(x)
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Cosine Pythagorean
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cos2(x)=1-sin2(x)
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Secant Pythagorean
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tan2(x)+1=sec2(x)
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Tangent Pythagorean
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tan2(x)=sec2(x)-1
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Secant-Tangent Pythagorean
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sec2(x)-tan2(x)=1
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Cosecant Pythagorean
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1+cot2(x)=csc2(x)
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Cotangent Pythagorean
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cot2(x)=csc2(x)-1
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Cosecant-Cotangent Pythagorean
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csc2(x)-cot2(x)=1
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sin(x+y)=sin(x)cos(y) + cos(x)sin(y)
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sin(x-y)=sin(x)cos(y) - cos(x)sin(y)
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cos(x+y)=cos(x)cos(y) - sin(x)sin(y)
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cos(x-y)=cos(x)cos(y) + sin(x)sin(y)
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sin(x)sin(y)=½(cos(x-y) - cos(x+y))
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cos(x)cos(y)=½(cos(x+y) + cos(x-y))
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sin(x)cosy)=½(sin(x+y) + sin(x-y))
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a2-u2
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Let u=asin(θ), du=acos(θ)dθ
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θ∈[-π/2, π/2]
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u2-a2
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Let u=asec(θ), du=asec(θ)tan(θ)dθ
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θ∈[0, π/2)
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u2+a2
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Let u=atan(θ), du=asec2(θ)dθ
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θ∈(-π/2, π/2)
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a2-b2u2
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Let u=(a/b)sin(θ), du=(a/b)cos(θ)dθ
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θ∈[-π/2, π/2]
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b2u2-a2
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Let u=(a/b)sec(θ), du=(a/b)sec(θ)tan(θ)dθ
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θ∈[0, π/2)
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b2u2+a2
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Let u=(a/b)tan(θ), du=(a/b)sec2(θ)dθ
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θ∈(-π/2, π/2)
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Sine Half-Angle
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sin(x/2)=√(½(1-cos(x)))
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Cosine Half-Angle
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cos(x/2)=√(½(1+cos(x)))
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Sine Power-Reducing
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sin2(x)=½(1-cos(2x))
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Cosine Power-Reducing
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cos2(x)=½(1+cos(2x))
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Sine Double-Angle
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sin(2x)=2sin(x)cos(x)
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Cosine Double-Angle 1
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cos(2x)=cos2(x)-sin2(x)
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Cosine Double-Angle 2
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cos(2x)=2cos2(x)-1
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Cosine Double-Angle 3
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cos(2x)=1-2sin2(x)
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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!
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