# The origins of $\operatorname{cis}(\theta)$

There is a abbreviation used in high school mathematics that is almost never seen outside of it: $$\operatorname{cis}(\theta) = \cos(\theta) + i \sin(\theta)$$, where cis stands for cosine + i sine.
As soon as students get into university Euler's formula $$e^{i\theta} = \cos(\theta) + i \sin(\theta)$$ is almost exclusively used as it makes a lot of things easier and more obvious. The reason Euler's formula is not used in high school is probably that it's not easy to prove or to develop an intuition for at that level.

My question is: does anyone know the origins and history of the abbreviation cis ?

The notation exists since a long time. It was used already by Irving Stringham in 'Uniplanar Algebra (1893).' This is claimed to be the earliest use on http://jeff560.tripod.com/trigonometry.html giving Cajori vol. 2, page 133 as reference.

In this book, Uniplanar Algebra, the notation is used first, as far as I can see, in chapter III (The algebra of complex quantities). It is introduced without much ado as a shorthand notation. It is said it should be read as "sector of $\theta$".

Cajori also mentions another early usage (thanks to Dag Oskar Madsen for the information!), namely, Harkness and Morley "Introduction to the theory of analytic functions (1898)". Also there it is introduced as shorthand, and the link to the exponential function that will be seen later is mentioned right away.

It seems the motivation is, as speculated in the question, to have some notation at hand when the notation via the exponential function is not yet available in a justified or motivated way.

• The quotation from Cajori: "Stringham denoted $\cos \beta + i \sin \beta$ by $\rm{cis} \beta$, a notation also used by Harkness and Morley." – Dag Oskar Madsen Sep 17 '14 at 14:40
• Thanks! I checked that reference now too and updated the answer. – quid Sep 17 '14 at 15:18
• Thanks @quid! I'd forgotten about the Jeff Miller pages. Also the links to archive.org are useful. – Simon Sep 18 '14 at 4:41
• It suddenly occurs to me one reason why cis persists in schools: many schools don't teach radians but do teach complex numbers. The notation $e^{i\theta}$ only makes sense if $\theta$ is in radians. – DavidButlerUofA Nov 9 '14 at 19:47
• That's an interesting observation @DavidButlerUofA that did not occur to me before. – quid Nov 9 '14 at 19:53