# What is the geometric interpretation of the value of the secant and cosecant of an angle?

I am confused about what is the geometric representation and interpretation of the secant and cosecant of an angle. I understand how to calculate them but I do not know what they mean, geometrically.

• I doubt there is really any.. they're just there to help you later on in future math courses; e.g. the derivative (calculus term) of the tangent function is secant squared. – pie314271 Dec 12 '16 at 23:48
• @pie314271 Do you still have doubts after seeing the diagrams below? – Théophile Dec 13 '16 at 0:16
• @Théophile: I was thinking that he was referring to the usages of secant/cosecant (e.g. $e^{ix}=\cos x+i\sin x$). Of course there's that, but based on the OP's response you're probably right. – pie314271 Dec 13 '16 at 0:34

In the usual terms or geometric representation of cos and sin on the unit circle in terms of some angle $\theta$ you can also get a 'geometric representation' of sec and cosec here also. See the image below. • I think the way of looking at it that is shown in Wikipedia's article on trigonometric functions is better. See my posted answer. – Michael Hardy Dec 13 '16 at 0:07
• @MichaelHardy Why do prefer the other diagram? – Théophile Dec 13 '16 at 0:15
• @Théophile : I may actually have to collect my thoughts on this in order to write as short an answer as the question deserves. – Michael Hardy Dec 13 '16 at 1:08
• @Théophile : One reason is that it allows a good explanation of why the tangent and secant are positive in certain quadrants and negative in certain other quadrants. $\qquad$ – Michael Hardy Dec 13 '16 at 1:09
• @MichaelHardy “If I had had more time, I would’ve written a shorter article.” — Mark Twain. – amd Dec 13 '16 at 7:11  ${{{{{{{{{{{{{{{{{{{{\qquad}}}}}}}}}}}}}}}}}}}}$

• This (rather than the other diagram) is the way I was taught long, long ago. – amd Dec 13 '16 at 7:12