In some contexts, the "cis" notation may serve the pedagogical purpose of emphasizing that one has not yet proved that this is an exponential function. In doing trigonometry without complex numbers, one may prove the two identities

Similarly in treating multiplication of complex numbers (with no involvement of trigonometry), one may observe that the real and imaginary parts of the product of "c"<sub>1</sub>&nbsp;+&nbsp;"is"<sub>1</sub> and "c"<sub>2</sub>&nbsp;+&nbsp;"is"<sub>2</sub> are respectively

Thus one sees this same pattern arising in two disparate contexts:


This coincidence can serve as a motivation for conjoining the two contexts and thereby discovering the trigonometric identity

and observing that this identity for cis of a sum is simpler than the identities for sin and cos of a sum. Having proved this identity, one can challenge the students to recall which familiar sort of function satisfies this same functional equation

The answer is exponential functions. That suggests that cis may be an exponential function

Then the question is: what is the base "b"? The definition of cis and the local behavior of sin and cos near zero suggest that

(where "dx" is an infinitesimal increment of "x"). Thus the rate of change at 0 is "i", so the base should be "e"<sup>"i"</sup>. Thus if this is an exponential function, then it must be

सममिति, अनतर एवं आवर्तन (Symmetry, shifts, and periodicity)
Pedagogy
Other sums of trigonometric functions
कोणों के योग और अन्तर की सर्वसमिकाएँ (Angle sum and difference identities)