Trig Cheat Sheet - Pauls Online Math Notes

1 2005 Paul Dawkins Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 02pq<< or 090q << . oppositesinhypotenuseq= hypotenusecscoppositeq= adjacentcoshypotenuseq= hypotenusesecadjacentq= oppositetanadjacentq= adjacentcotoppositeq= Unit circle definition For this definition q is any angle. sin1yyq== 1cscyq= cos1xxq== 1secxq= tanyxq= cotxyq= Facts and Properties Domain The domain is all the values of q that can be plugged into the function. sinq , q can be any angle cosq, q can be any angle tanq, 1,0,1,2,2nnqp += K cscq, ,0,1,2,nnqp = K secq, 1,0,1,2,2nnqp += K cotq, ,0,1,2,nnqp = K Range The range is all possible values to get out of the function. 1sin1q- csc1andcsc1qq - 1cos1q- sec1andsec1qq - tanq- << cotq- << Period The period of a function is the number, T, such that ()()fTfqq+=.

2 So, if w is a fixed number and q is any angle we have the following periods. ()sinwq 2 Tpw= ()coswq 2 Tpw= ()tanwq Tpw= ()cscwq 2 Tpw= ()secwq 2 Tpw= ()cotwq Tpw= q adjacent opposite hypotenuse x y (),xyqx y 1 2005 Paul Dawkins Formulas and Identities Tangent and Cotangent Identities sincostancotcossinqqqqqq== Reciprocal Identities 11cscsinsincsc11seccoscossec11cottantanc otqqqqqqqqqqqq====== Pythagorean Identities 222222sincos1tan1sec1cotcscqqqqqq+=+=+= Even/Odd Formulas ()()()()()()sinsincsccsccoscossecsectant ancotcotqqqqqqqqqqqq-=--=--=-=-=--=- Periodic Formulas If n is an integer. ()()()()()()sin2sincsc2csccos2cossec2sec tantancotcotnnnnnnqpqqpqqpqqpqqpqqpq+=+= +=+=+=+=Double Angle Formulas ()()()22222sin22sincoscos2cossin2cos112s in2tantan21tanqqqqqqqqqqq==-=-=-=- Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then 180and 180180txttxxppp=fi==Half Angle Formulas (alternate form) ()()()()()()2221cos1sinsin1cos22221cos1c oscos1cos22221cos21costantan21cos1cos2qq qqqqqqqqqqqq-= =-+= =+--= =++Sum and Difference Formulas ()()()sinsincoscossincoscoscossinsintant antan1tantanababababababababab = = =mm Product to Sum Formulas ()()()()()()()()

3 1sinsincoscos21coscoscoscos21sincossinsi n21cossinsinsin2abababababababababababab =--+ =-++ =++- =+-- Sum to Product Formulas sinsin2sincos22sinsin2cossin22coscos2cos cos22coscos2sinsin22abababababababababab abab+- += +- -= +- += +- -=- Cofunction Formulas sincoscossin22cscsecseccsc22tancotcottan 22ppqqqqppqqqqppqqqq -=-= -=-= -=-= 2005 Paul Dawkins Unit Circle For any ordered pair on the unit circle (),xy : cosxq= and sinyq= Example 5153cossin3232pp ==- 3p 4p 6p 22,22 31,22 13,22 60 45 30 23p 34p 56p 76p 54p 43p 116p 74p 53p 2p p 32p 0 2p 13,22 - 22,22 - 31,22 - 31,22 -- 22,22 -- 13,22 -- 31,22 - 22,22 - 13,22 - ()0,1 ()0,1- ()1,0- 90 120 135 150 180 210 225 240 270 300 315 330 360 0 x ()1,0 y 2005 Paul Dawkins Inverse Trig Functions Definition 111sin is equivalent to sincos is equivalent to costan is equivalent to tanyxxyyxxyyxxy---====== Domain and Range Function Domain Range 1sinyx-= 11x- 22ypp- 1cosyx-= 11x- 0yp 1tanyx-= x- << 22ypp-<< Inverse Properties ()()()()()()()()()()()()

4 111111coscoscoscossinsinsinsintantantant anxxxxxxqqqqqq------====== Alternate Notation 111sinarcsincosarccostanarctanxxxxxx---= == Law of Sines, Cosines and Tangents Law of Sines sinsinsinabcabg== Law of Cosines 2222222222cos2cos2cosabcbcbacaccabababg= +-=+-=+- Mollweide s Formula ()1212cossinabcabg-+= Law of Tangents ()()()()()()121212121212tantantantantant anababbcbcacacababbgbgagag--=++--=++--=+ +c a b a b g