It is given that the complex number p = cos 3x + isin 3x while q = cos 4x + isin3x.?

Question

Express the complex number p+q in x+iy and exponential form. Help!!! I can't work this out because the angle for q is differrent under sin and cos.

Answer 1

Feel that question should read:-
p = cos 3x + i sin 3x
q = cos 4x + i sin 4x

p + q
= (cos 3x + cos 4x) + i (sin 3x + sin 4x)
= 2 cos(7x/2)cos(x/2) + i (sin(7x/2) cos (x/2))
= a + i b
= e^(i θ) where θ = tan^(-1) (b/a)

θ = tan^(-1) tan (7x/2)
θ = 7x/2

p + q = e^(i 7x/2)
p + q = cos (7x/2) + i sin (7x/2)

Answer 2

p+q= is a resultant of two complex numbers so their sum is the diagonal of parallelogram produced from them on Argand diagram ,then its amplitude is 3.5X and its modulus is r= 2cosx/2
then p+q= 2cos x/2 ( cos3.5x +isin3.5x)

and its exponential form is 2cos x/2 e^i 3.5xπ/180

Answer 3

p+q = (cos3x +cos4x) +2isin3x
now p+q =r*e^(i w)
where r =sqrt( (cos3x +cos4x)^2 + (2sin3x)^2)
and w =tan^-1(2sin3x/(cos3x +cos4x))