How should trig functions be pronounced?

A good reference for the pronunciation of mathematical terms and notation is the Handbook for Spoken Mathematics by Lawrence A. Chang. This handbook is, perhaps, a bit pedantic in places (one of the primary target audiences is computer scientists and engineers who are interested in voice synthesis, though I would also recommend the book for instructors of students with poor vision or blindness), but it typically gives a good place to start.

Paraphrasing Chang:

\begin{align} &\sin \theta &&\text{sine [of] theta} \\ &\cos \theta &&\text{co sine [of] theta} \\ &\tan \theta &&\text{tangent [of] theta} \\ &\csc \theta &&\text{co secant [of] theta} \\ &\sec \theta &&\text{secant [of] theta} \\ &\cot \theta &&\text{co tangent [of] theta} \\\\ &\sinh \theta &&\text{hyperbolic sine theta –OR– sinch theta} \\ &\cosh \theta &&\text{hyperbolic co sine theta –OR– cosh theta} \\ &\tanh \theta &&\text{hyperbolic tangent theta –OR– tange theta} \\ &\operatorname{csch} \theta &&\text{hyperbolic co secant theta} \\ &\operatorname{sech} \theta &&\text{hyperbolic secant theta} \\ &\coth \theta &&\text{hyperbolic co tangent theta} \\\\ &\operatorname{arc\langle trig\rangle} x && \}\qquad\text{arc $\langle$trig$\rangle$ [of] eks} \\ &\ \ \ \ \text{OR} && \}\qquad\text{inverse $\langle$trig$\rangle$ [of] eks}\\ &\operatorname{\langle trig\rangle}^{-1} x && \}\qquad\text{anti $\langle$trig$\rangle$ [of] eks} && (\ast) \end{align}

Notes

• At ($\ast$), the notation $\langle$trig$\rangle$ denotes any one of the trigonometric or hyperbolic trigonometric functions above. Note that Chang also suggests using "$\langle$trig$\rangle$ to the minus one", though I would find this inadvisable, as this might be confused for the reciprocal.

• Chang seems to be ambivalent about the inclusion of the word "of" in the notation. Personally, when speaking to students, I tend to leave it in, as it emphasizes that these things are functions which take an argument. In the same way, I always pronounce $f(x)$ as "eff of eks", and never "eff eks".

• I have heard "cahs theta", "tan theta" and "seek theta" in the wild for the cosine, tangent, and secant functions, respectively. There is nothing wrong with either, though I tend to avoid them for the sake of clarity.

• In a similar vein, "sine aitch", "cahs aitch", and "tan aitch" are common spoken shortcuts for the hyperbolic sine, cosine, and tangent, respectively. Given that "hyperbolic sine" is something of a pain to say and that "sinch" makes my ears hurt, "sine aitch" seems like a reasonable compromise. Of course, I have now gone way over into the land of opinion.

• As pointed out in a chat comment, the abbreviated versions of these functions may have regionally-variant pronunciations. For example, $\sinh$ is generally pronounced "sinch" in American settings, but is apparently pronounced "shine" in British English. This argues in favor of using the entire phrase, i.e. "hyperbolic sine".

For sine and cosine and the inverse functions that begin with "arc" you say the full name. For tangent and arctangent you usually omit the "gent" at the end, but you need not.

Secant, cosecant and cotangent are less common. Use the full names.