A polarizer passes only the field component along its transmission axis, so the surviving amplitude is E0 * cos(theta). This is an ideal projection: the polarizer is perfect, the input field is a single clean vector, and the axis angle is known exactly. Real polarizers leak, absorb, and misalign, so cos(theta) is a model for the stated inputs, not a measurement.
A linear polarizer keeps the requested axis field component and blocks the orthogonal one. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
An ideal polarizer keeps the axis component
For axis mode, the incoming 10 volts per metre splits with factor 3/5 for the passed component and orthogonal remainder.