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Trigonometric Addition Identities OP 1 OQ cos A PQ sin A OR cos A cos B QR cos A sin B QS sin A cos B PS sin A sin B P cos( A B) OT OR PS cos A cos B sin A sin B sin A 1 UA cos sin( A B) PT QS QR S B Q A sin A cos B cos A sin B B O T R The tan addition formula Q What is tan in terms of sin and cos ? sin A tan cos Q What is tan( A B), and how are you going to continue? A tan( A B) sin( A B) sin A cos B cos A sin B cos( A B) cos A cos B sin A sin B Q How are you going to express this in terms of tan A and tan B ? A Divide top and bottom by cos A cos B sin A cos B cos A sin B cos A cos B cos A cos B tan A tan B tan( A B) cos A cos B sin A sin B 1 tan A tan B cos A cos B cos A cos B The circular definitions of trigonometric functions The right angled triangle definition R X cos , so X R cos , R Y sin , so Y R sin R Y X 1 Scale the triangle, to make the hypoteneuse 1 x cos y sin y sin tan x cos y x P(x,y) 1 Place this triangle in a unit radius circle, with centre at the origin We now turn things round and DEFINE ( cos ,sin ) as the coordinates of P, where OP makes angle with the x axis. y O x Deductions from the circular definition To show the angles, Polar coordinates have been used P1[1, ] means OP1 1, and OP1 makes angle with the P2 [1,180] P1[1,] (cos ,sin ) (cos ,sin ) positive x axis You should be able to see how to write the cos or sin of any of , 180 , 360 in terms of the cos or sin of P3 [1,180] P4 [1,360 ] (cos ,sin ) (cos ,sin ) cos(180 ) cos cos(180 ) cos cos(360 ) cos sin(180 ) sin sin(180 ) sin sin(360 ) sin cos( ) cos(360 ) cos sin( ) sin(360 ) sin Trigonometric Subtraction Identities We could modify the addition identities argument by using the cicular definitions, and replacing lengths by signed lengths, and will assume this can be done, but not here. cos( A B) cos( A ( B)) cos A cos( B) sin A sin( B) cos A cos B sin A sin B sin( A B) sin( A ( B)) sin A cos( B) cos A sin( B) sin A cos B cos A sin B