Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that Unit circle definition p For this definition q is any angle. 0 <<q or 0°<q <°90 . 2 y ( xy, ) 1 hypotenuse y q opposite x x q adjacent opposite hypotenuse y 1 sinq = cscq = sinq ==y cscq = hypotenuse opposite 1 y adjacent hypotenuse x 1 cosq = secq = cosq ==x secq = hypotenuse adjacent 1 x opposite adjacent y x tanq = cotq = tanq = cotq = adjacent opposite x y Facts and Properties Domain The domain is all the values of q that Period can be plugged into the function. The period of a function is the number, T, such that f(qq+=Tf) ( ) . So, if w sinq , q can be any angle is a fixed number and q is any angle we cosq , q can be any angle have the following periods. æö1 tanq , qp¹ç÷nn+,=0,±±1,2,K èø2 2p sin (wq) ® T = cscq , qp¹nn,=0,±±1,2,K w æö1 2p secq , qp¹ç÷nn+,=0,±±1,2,K cos(wq) ® T = èø2 w p cotq , qp¹nn,=0,±±1,2,K tan (wq) ® T = w Range 2p csc(wq) ® T = The range is all possible values to get w out of the function. 2p sec(wq) ® T = -1££sin1q cscqq³1andcsc1£- w -1££cos1q secqq³1andsec1£- p cot (wq) ® T = -¥<tanq <¥ -¥<cotq <¥ w © 2005 Paul Dawkins Formulas and Identities Tangent and Cotangent Identities Half Angle Formulas (alternate form) sinqqcos qq1-cos1 tanqq==cot sin=±sin2 qq=-(1cos2( )) cosqqsin 222 Reciprocal Identities qq1+cos1 11cos=±cos2 qq=+(1cos2( )) cscqq==sin 222 sinqqcsc 11qq1-cos 2 1-cos2( q ) secqq==cos tan=±=tan q cosqqsec 21++cosqq1cos2( ) 11Sum and Difference Formulas cotqq==tan tanqqcot sin(a±b) =±sinacosbcosabsin Pythagorean Identities cos(a±=b) cosacosbmsinabsin sin22qq+=cos1 tanab±tan tan22qq+=1sec tan (ab±=) 1m tanabtan 1+=cot22qqcsc Product to Sum Formulas Even/Odd Formulas 1 sinasinb=ëûéùcos(a-b) -+cos(ab) sin(-q) =-sinqcsc(-qq) =-csc 2 1 cos(-q) =cosqsec(-=qq) sec cosacosb=éùcos(a-b) ++cos(ab) 2 ëû tan(-q) =-tanqcot(-qq) =-cot 1 sinacosb=ëûéùsin(a+b) +-sin (ab) Periodic Formulas 2 If n is an integer. 1 cosasinb=ëûéùsin(a+b) --sin (ab) sin(q+2pnn) =sinqcsc(q+=2pq) csc 2 Sum to Product Formulas cos(q+2pnn) =cosqsec(q+=2pq) sec æa+-böæöab tanq+pnn=tanqcotq+=pqcot sinab+=sin2sinç÷cosç÷ ( ) ( ) è22øèø Double Angle Formulas æa+-böæöab sinab-=sin2cosç÷sin ç÷ sin(2q) = 2sinqqcos è22øèø cos(2q) =-cos22qqsin æa+-böæöab cosab+=cos2cosç÷cosç÷ 2 22 =-2cos1q èøèø 2 æa+-böæöab =-12sin q cosab-cos=-2sinç÷sin ç÷ è22øèø 2tanq tan2( q ) = Cofunction Formulas 1-tan2q æppöæö Degrees to Radians Formulas sinç-q÷=cosqcosç÷-=qqsin 22 If x is an angle in degrees and t is an èøèø æppöæö angle in radians then cscç-q÷=secqsecç÷-=qqcsc pptxt180 è22øèø =Þtx==and æppöæö 180x180 p tanç-q÷=cotqcotç÷-=qqtan è22øèø © 2005 Paul Dawkins Unit Circle y (0,1) p æö13 æö13 ç÷, ç÷- , 2 èø22 èø22 p æö22 2p 90° ç÷, æö22 3 ç÷22 ç÷- , èø 22 3 p èø 120° 60° 3p æö31 4 ç÷, ç÷22 æö31 4 èø ç÷- , 135° 45° p èø22 5p 6 6 30° 150° (-1,0) p 180° 0° 0 (1,0) 360° 2p x 210° 7p 330° 11p 6 225° 6 æö31 æö31 ç÷--, 315° ç÷,- èø22 5p èø22 240° 7p 4 300° æö22 4p 270° ç÷--, 5p 4 æö22 èø22 3 3p ç÷,- 3 èø22 2 æö13 æö13 ç÷--, ç÷,- èø22 èø22 (0,1- ) For any ordered pair on the unit circle ( xy, ) : cosq = x and sinq = y Example æ5ppö1æö53 cosç÷=sin ç÷=- è3ø2èø32 © 2005 Paul Dawkins Inverse Trig Functions Definition Inverse Properties y==sin-1 x is equivalent to xysin cos(cos--11( xx)) ==cos(cos(qq)) -1 y==cosx is equivalent to xycos sin(sin--11()xx)==sin(sin ()qq) y==tan-1 x is equivalent to xytan tantan--11()xx==tan(tan ()qq) ( ) Domain and Range Function Domain Range Alternate Notation pp sin-1 xx= arcsin yx= sin-1 -11££x -££y 22 cos-1 xx= arccos -1 yx= cos -11££x 0 ££y p tan-1 xx= arctan pp yx= tan -1 -¥<x <¥ -<<y 22 Law of Sines, Cosines and Tangents c b a a g b Law of Sines Law of Tangents sinasinbgsin ab- tan 1 (ab- ) == = 2 abc 1 ab++tan 2 (ab) Law of Cosines 1 bc-tan 2 (bg-) 222 = a=b+-c2bc cosa 1 bc++tan 2 (bg) 222 b=a+-c2ac cos b 1 ac-tan 2 (ag-) 222 = c=a+-b2ab cosg 1 ac++tan 2 (ag) Mollweide’s Formula ab+ cos 1 (ab- ) = 2 1 c sin 2 g © 2005 Paul Dawkins .