exponential presentation of a complex number with Sine and Cosine multipliers

Question

I know $e^{i\theta} =\cos\theta+ i\sin\theta$, but is there any exponential representation for $a \cos\theta + ib \sin\theta$ as well ?

Answer 1

Since $e^{-i\theta}=\cos\theta-i\sin\theta$ we notice

$$k_1e^{i\theta}+k_2e^{-i\theta}=(k_1+k_2)\cos\theta+i(k_1-k_2)\sin\theta$$

Compare coefficient with the form desired, we get $$\frac{a+b}{2}e^{i\theta}+\frac{a-b}{2}e^{-i\theta}=a\cos\theta+ib\sin\theta$$