# Why does the complex representation of a voltage $V\cdot\cos(\omega t + \phi)$ use the cosine function?

A voltage across a circuit component at time $t$ can be calculated by $V\cdot\sin(\omega t + \phi)$.

However, in the section of my book that introduces complex numbers to represent voltages, it is stated that a voltage $V\cdot\cos(\omega t)$ can be represented by $V\cdot e^{j\theta}$ where $e^{j\theta} = \cos(\theta) + j\sin(\theta)$.

My confusion arises from the fact that the actual voltage is its amplitude times the cosine function and not the sine function since it is a sinusoid. Is this an error or I am just confused?

• Be aware that $\cos(\theta) = \sin(\theta + \frac{\pi}{2})$ so the distinction you seem to be drawing doesn't exist. A sinusoidal function of time can equally be expressed as $f(t) = \cos(\omega t + \phi) = \sin(\omega t + \phi + \frac{\pi}{2}) = \sin(\omega t + \phi')$ – Alfred Centauri Jan 12 '15 at 3:32