Trig Cheat Sheet Formulas and Identities Tangent and Cotangent Identities Half Angle Formulas (alternate form) sinqqcos qq1-cos1 Definition of the Trig Functions tanqq==cot sin=±sin2 qq=-(1cos2( )) Right triangle definition cosqqsin 222 Reciprocal Identities For this definition we assume that Unit circle definition qq1+cos1 p For this definition q is any angle. 11cos=±cos2 qq=+(1cos2( )) 0 <<q or 0°<q <°90 . cscqq==sin 222 2 y sinqqcsc 1-cos2q 11qq1-cos 2 ( ) secqq==cos tan=±=tan q ( xy, ) cosqqsec 21++cosqq1cos2( ) 1 11Sum and Difference Formulas hypotenuse y q cotqq==tan sina±b=±sinacosbcosabsin opposite x tanqqcot ( ) x Pythagorean Identities cos(a±=b) cosacosbmsinabsin q sin22qq+=cos1 tanab±tan adjacent 22 tan (ab±=) tanqq+=1sec 1tanabtan 22 m opposite hypotenuse y 1 1+=cotqqcsc Product to Sum Formulas sinq = cscq = sinq ==y cscq = hypotenuse opposite 1 y Even/Odd Formulas 1 sinasinb=ëûéùcos(a-b) -+cos(ab) adjacent hypotenuse x 1 sin(-q) =-sinqcsc(-qq) =-csc 2 cosq = secq = cosq ==x secq = hypotenuse adjacent 1 x 1 cos(-q) =cosqsec(-=qq) sec cosacosb=ëûéùcos(a-b) ++cos(ab) opposite adjacent y x 2 tanq = cotq = tanq = cotq = tan(-q) =-tanqcot(-qq) =-cot adjacent opposite x y 1 sinacosb=ëûéùsin(a+b) +-sin(ab) Periodic Formulas 2 Facts and Properties If n is an integer. 1 cosasinb=ëûéùsin(a+b) --sin(ab) Domain sin(q+2pnn) =sinqcsc(q+=2pq) csc 2 The domain is all the values of q that Period Sum to Product Formulas cos(q+2pnn) =cosqsec(q+=2pq) sec can be plugged into the function. The period of a function is the number, æa+-böæöab tanq+pnn=tanqcotq+=pqcot sinab+=sin2sinç÷cosç÷ T, such that f(qq+=Tf) ( ) . So, if w ( ) ( ) è22øèø sinq , q can be any angle Double Angle Formulas is a fixed number and q is any angle we æa+-böæöab cosq , q can be any angle have the following periods. sinab-=sin2cosç÷sin ç÷ sin(2q) =2sinqqcos è22øèø æö1 tanq , qp¹ç÷nn+,=0,±±1,2,K 22 èø2 2p cos(2q) =-cosqqsin æa+-böæöab sin(wq) ® T = cosab+=cos2cosç÷cosç÷ 2 22 cscq , qp¹nn,=0,±±1,2,K w =-2cos1q èøèø æö1 2p 2 æa+-böæöab secq , qp¹ç÷nn+,=0,±±1,2,K cos (wq) ® T = =-12sin q cosab-cos=-2sinç÷sinç÷ èø2 w è22øèø 2tanq qp¹nn,=0,±±1,2, p tan2q= Cofunction Formulas cotq , K tan (wq) ® T = ( ) 2 w 1-tan q æppöæö Range 2p Degrees to Radians Formulas sinç-q÷=cosqcosç÷-=qqsin csc(wq) ® T = è22øèø The range is all possible values to get w If x is an angle in degrees and t is an æppöæö out of the function. 2p angle in radians then cscç-q÷=secqsecç÷-=qqcsc sec(wq) ® T = è22øèø -1££sin1q cscqq³1andcsc1£- w pptxt180 =Þtx==and æppöæö -1££cos1q secqq³1andsec1£- p 180x180 p tanç-q÷=cotqcotç÷-=qqtan cot (wq) ® T = è22øèø -¥<tanq <¥ -¥<cotq <¥ w © 2005 Paul Dawkins © 2005 Paul Dawkins Unit Circle Inverse Trig Functions Definition Inverse Properties y -1 --11 (0,1) y==sinx is equivalent to xysin cos(cos( xx)) ==cos(cos(qq)) -1 p æö13 --11 ç÷, y==cosx is equivalent to xycos sinsinxx==sinsin qq æö13 ç÷22 ( ()) ( ()) ç÷-, 2 èø -1 èø22 y==tanx is equivalent to xytan --11 p æö22 tan(tan()xx)==tan(tan ()qq) 2p 90° ç÷, æö22 3 ç÷22 ç÷- , èø 22 3 p Domain and Range èø 120° 60° Alternate Notation 3p æö31 Function Domain Range 4 ç÷, -1 ç÷22 pp sinxx= arcsin æö31 4 èø yx=sin-1 -11££x -££y ç÷- , 135° 45° p -1 èø22 5p 22 cosxx= arccos 6 -1 6 30° yx=cos -11££x 0 ££y p tan-1 xx= arctan 150° pp yx=tan -1 -¥<x <¥ -<<y 22 (-1,0) p 180° 0° 0 (1,0) Law of Sines, Cosines and Tangents 360° 2p x c b a 210° 7p 330° 11p 6 225° 6 æö31 æö31 a g ç÷--, 315° ç÷,- èø22 5p èø22 240° 300° 7p 4 270° æö22 4p b ç÷--, 5p 4 æö22 èø22 ç÷,- 3 3p 22 3 èø Law of Sines Law of Tangents æö13 2 --, æö13 1 ç÷ ç÷,- sinasinbgsin ab- tan 2 (ab-) èø22 èø22 == = abc 1 (0,1- ) ab++tan 2 (ab) Law of Cosines 1 bc-tan 2 (bg-) 222 = a=b+-c2bc cosa 1 bc++tan 2 (bg) 222 For any ordered pair on the unit circle ( xy, ) : cosq = x and sinq = y b=a+-c2ac cos b 1 ac-tan 2 (ag-) 222 = c=a+-b2ab cosg ac++tan 1 (ag) Example 2 Mollweide’s Formula æ5ppö1æö53 1 cosç÷=sinç÷=- ab+ cos 2 (ab-) è3ø2èø32 = c sin 1 g 2 © 2005 Paul Dawkins © 2005 Paul Dawkins .