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Unformatted text preview: ts more s imply . Some ex amples of this are s hown by
s hifting func tions round by π/2, π and 2π radians . Bec aus e the periods of thes e func tions are either π or 2π, there are c as es where the new func tion is ex ac tly the s ame as the old func tion
without the s hift.
Shift by π/2 Shift by π
Pe riod for ta n a nd cot en.wikipedia.or g /wiki/List_of_tr ig onometr ic_identities Shift by 2π
Pe riod for sin, cos, csc a nd se c 2/12 12/7/13 List of tr ig onometr ic identities - Wikipedia, the fr ee encyclopedia Angle sum and dif f erence identities  See als o: #Produc t-to-s um and s um-to-produc t identities
Thes e are als o k nown as the addition and s ub trac tion theorems or formulae. They were originally es tablis hed by the 10th c entury Pers ian
mathematic ian Abū al-W afā' Būz jānī. One method of proving thes e identities is to apply Euler's formula. The us e of the s y mbols and
des c ribed in the artic le plus -minus s ign.
For the angle addition diagram for the s ine and c os ine, the line in bold with the 1 on it is of length 1. It is the hy potenus e of a right angle triangle
with angle β whic h gives the s in β and c os β. The c os β line is the hy potenus e of a right angle triangle with angle α s o it has s ides s in α and c os
α both multiplied by c os β. This is the s ame for the s in β line. The original line is als o the hy potenus e of a right angle triangle with angle α + β, the
oppos ite s ide is the s in(α + β) line up from the origin and the adjac ent s ide is the c os (α + β) s egment going horiz ontally from the top left.
Overall the diagram c an be us ed to s how the s ine and c os ine of s um identities bec aus e the oppos ite s ides of the rec tangle are equal.
S i ne  Cosine 
 Ta nge nt  Arcsine  Arccosine
 Arcta nge nt Matrix form Illus tr ation of angle addition f or mulae f or
the s ine and c os ine. Emphas iz ed s egment is
of unit length.  See als o: matrix multiplic ation
The s um and differenc e formulae for s ine and c os ine c an be written in matrix form as :
Illus tr ation of the angle addition f or mula
f or the tangent. Emphas iz ed s egments ar e of
unit length. This s hows that thes e matric es form a repres entation of the rotation group in the plane (tec hnic ally , the s pec ial orthogonal group SO(2)), s inc e the c ompos ition law is fulfilled: s ubs equent
multiplic ations of a vec tor with thes e two matric es y ields the s ame re...
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This document was uploaded on 02/04/2014.
- Spring '14