 When you add a cosine wave and a sine wave with different amplitudes but the same frequency together, you get a sinusoid with the same frequency but a different amplitude and phase. You've probably learned this at school as the identity A times the cosine of x plus B times the sine of x equals C times the cosine of x minus alpha. The amplitude of the new wave C is the Pythagorean combination of A and B, and the phase shift alpha is the inverse tangent of B divided by A. This little identity is very powerful because it means that by simply modifying the amplitudes of locally generated cosine and sine waves, it can be used to transmit data via radio waves using phase or frequency modulation. Alternatively, by multiplying a signal by locally generated cosine and sine waves at many different frequencies, the properties of that signal can be analyzed using the Fourier transform.