finding the sinusoidal values of a complex current or voltage
The "big I" notation is called "phasor notation", which is a shorthand notation that only represents the phase and magnitude of a signal, while ignoring its frequency. It uses rectangular coordinates, with a real component and an imaginary component. The general form is

$$I = Real + j\cdot Imaginary$$

The "i(t)" notation is a direct representation of the same signal in the time domain, but now the phase and magnitude are given using polar coordinates. The general form is

$$i(t) = A \sin(\omega t + \theta)$$

where A is the amplitude, &omega; is the radian frequency and &theta; is the phase angle.

You can look up "rectangular to polar" and "polar to rectangular" coordinate conversions to convert between the two notations.