Question

How do you express the complex number in trigonometric form `2(cos 90° + i sin 90°)`?

Answer 1

In a sense, it's already in trigonometric form. Other common ways to write it, which may be what you are after, are `2\mbox{cis}(90^{circ})` and `2e^{i*90^{circ})=2e^{i*pi/2}`.

Explanation:

The second form follows from Euler's formula: `e^{i theta}=cos(theta)+i sin(theta)`.

The "cis" form of the answer is just another way to write it (`mbox{cis}(theta)` is a symbol that is, by definition, equal to `cos(theta)+i sin(theta)`).

All of this can also be thought of in terms of polar coordinates in the complex plane. The polar coordinates of the complex number `2(cos(90^{circ})+i sin(90^{circ}))=2i` are `(r,theta)=(2,90^{circ})` (its rectangular coordinates are `(x,y)=(0,2)`).