Magnitude via Vector Sum

It often happens that we can't directly measure the amplitude and phase angle of a sinusoid, but instead we measure its sine and cosine "component" amplitudes. In the sine wave clock example we measured the sine component of the clock face (the height of the hand's pointer) to get our sine waveform. The cosine component is measured similarly, as the length of the "shadow" of the hand on the horizontal axis:

We can thus find the true amplitude or "magnitude" as the hypotenuse of this right triangle by taking the square root of the sum of the squares of the sine and cosine components. This is called the "vector sum" of the components. Similarly, we can find the phase as the angle whose tangent is equal to the sine component divided by the cosine component.

The sine and cosine components are often referred to as the "in-phase" and "quadrature" components, especially in discussions of older analog methods. Engineering math types prefer to use "real" for the cosine component and "imaginary" for the sine component, corresponding to a wonderfully intimidating exponential expression called "complex notation", perfect for casual dining conversation and impressing uninitiated audiences.

See also Sine Wave Basics, Sine Wave Phase, Making Waves via Sine Wave Synthesis