Math 1316 Formulas Half-Angle Formulas Sum

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