D e fi n i t i o n of the Six TrigonometricFunctions ( 0 ,1 ) Riglr tt r ianglectefnition.r,*'here 0 < 0 < rr/2. opposite l ry p . tan0: Arljucerrt 2 Iryp. sing=-g csc6: nyp. opp. adi. hVg = cos 0 ODD, -t- aoJ, r\ 2Lr v 5 /1 sec0= cot 0 : adj , adj . opp. rrt Jlt \-f,T) rrt 1r \T,i) T r Jl, Jit \-'ftf) r ,ft l\ \--Ttl) 600 45' 300 (-1,0) Circular function definitions, where 0 is any angle. (1,0) YT 330o sin0=1 csc0:ry cos0:tan0:1 XT rx VX xy scc0=cot0:- r \ rt 2 ' 225' L4 z4o" lr -1E1-, r 3150 (+,-L) 2/ rrt e+,-*) /\ - r2 , - f \z t \Tt-TJ --- rtt r\ 2rr - / 25 r) (0,_l) Double-AngleFormulas Reciprocalldentities .11 slnu: csc u csc u ll s ln u secu:- cos u s e cu I cot u I cotu:tun u tanu:- Tangentand Cotangentldentities tatru:- s in u c os u cosu cOtu s tn u tilrt'u Power-ReducingFormulas l-cos2u .1 sln'u -2 1 * c os 2u cos." u = --;^ : tan" u I * cotz u = csczu Cofunctionldentities . lrr \ l r - u\ : s i n u ttn\z - u) : cosu cost; ,| \r, / lrr - \ l rr \ .t.\t ,,) : secrl t t n ( t - u i = cot tl / lrr \ ln \ t..\t - u) - cscu c o t \ z - u :tanu ) NegativeAngle ldentities sin(-u) : -sin u csc(-u) : -csc u s e c ( - u ) = s e cu tirn2u:.-- I 2 c o s 2u - t - l ' - 2 s i n 2u L Pythagoreanldentities sinZu*cos2u:l I -f tan2u : sec2u sin2u:2sinucosu c o s2 u : c o s 2u - s i n 2u : 2tanu cos(-u) = cosu tan(-u) = -tan u cot(-u) = -cot u I -cos 2u T+;;t Sum-to-ProductForrnulas sinu t sinv = 2',"(+) ."r(+) \4/ \2/ sinu - siny = 2."'(+)''"(?) cosu -r-cosv = z'"'(+) ."'(+)L / \ \z/ cosu - cosv: -zt'"(+) \ Z/ t,.(+) \ L Formulas Product-to-Sum I s i n u s i n v : r l c o s ( u - v ) - cos(u-Fv)l Surn and DifferenceFormulas cosu cosv = jt.ort,r-v) * c o s ( u * v ) l si n( ut v ) : s in u c o s v -t c o s u s i n v c o s ( u t v ) : c o su c o sv T s i n u s i n v sinucosv:jtrintu*v)t tall(uiv) : tanu+tunv ;-=-_ lTtanutanv si n(u- v)l c o su s i n v - - j t r , n t u * u ) - s i n ( u- v ) l /