while the same in polar coordinates gives:
equating the real and imaginary parts of the two versions of this one product then painlessly yields the standard angle sum formulas of trigonometry:
Alternatively, the polar form for complex multiplication can be derived using these trigonometric formulas. Indeed, the rule that we have stated here, without proof, for multiplication in polar form is usuallyrst derived from the rectangular form by using trigonometric formulas.
Much more comes quite easily now as the use of complex numbers reveals a connection between the exponential or power function,and the seemingly unrelated trigonometric functions. Without passing through the portal offered by the square root of minus one, the connection may be glimpsed, but not understood. The so-called hyperbolic functions arise from taking what are known as the even and odd parts of the exponential function. To every trigonometric identity there corresponds one of identical form,except perhaps for sign, involving these hyperbolic functions. This can be veried easily in any particular case, but then the question remains as to why it should happen at all. Why should the behaviour of one class of functions be so closely mirrored in another class, dened in so different a manner, and of such different character? Resolution of the mystery is by way of the formula eiθ=cosθ+i sinθ, which shows that the exponential and trigonometric functions are intimately linked, but only through use of the imaginary unit i. Once this is revealed(for it is surprising and is by no means obvious), it becomes clear that results along the lines described are inevitable through performing calculations using the two alternative representations offered by this equation and then equating real and imaginary parts. Without the formula, however, it all remains a mystery.