Math 1316 Formulas Half-Angle Formulas r u 1 − cos u sin = ± 2 2 r u 1 + cos u cos = ± 2 2 u 1 − cos u sin u tan = = 2 sin u 1 + cos u Sum and Difference Formulas sin(u+v) = sin u cos v+ cos u sin v cos(u+v) = cos u cos v+ sin u sin v tan u+ tan v tan(u+v) = 1+ tan u tan v Sum-to-Product Formulas u−v sin u + sin v = 2 sin( u+v 2 ) cos( 2 ) u−v sin u − sin v = 2 cos( u+v 2 ) sin( 2 ) u−v cos u + cos v = 2 cos( u+v 2 ) cos( 2 ) u−v cos u − cos v = −2 sin( u+v 2 ) sin( 2 ) Product-to-Sum Formulas sin u sin v = 12 [cos(u − v) − cos(u + v)] cos u cos v = 12 [cos(u − v) + cos(u + v)] sin u cos v = 12 [sin(u + v) + sin(u − v)] cos u sin v = 12 [sin(u + v) − sin(u − v)] Product and Quotient Formulas If z1 = r1 (cos θ1 + i sin θ1 ) and z2 = r2 (cos θ2 + i sin θ2 ), then z1 z2 = r1 r2 [cos(θ1 + θ2 ) + i sin(θ1 + θ2 )] z1 r1 z2 = r2 [cos(θ1 − θ2 ) + i sin(θ1 − θ2 )], z2 6= 0 Power and Root Formulas If z = r(cos θ + i sin θ), then z n = rn (cos nθ + i sin nθ) If z = r(cos θ + i sin θ), then the nth roots of z are: √ n r(cos θ+2πk + i sin θ+2πk n n ) where k = 0, 1, 2, . . . , n − 1. 1