# 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. Dec 12, 2016 at 23:48
• @pie314271 Do you still have doubts after seeing the diagrams below? Dec 13, 2016 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. Dec 13, 2016 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. Dec 13, 2016 at 0:07
• @MichaelHardy Why do prefer the other diagram? Dec 13, 2016 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. Dec 13, 2016 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. $\qquad$ Dec 13, 2016 at 1:09
• @MichaelHardy “If I had had more time, I would’ve written a shorter article.” — Mark Twain.
– amd
Dec 13, 2016 at 7:11  ${{{{{{{{{{{{{{{{{{{{\qquad}}}}}}}}}}}}}}}}}}}}$

• This (rather than the other diagram) is the way I was taught long, long ago.
– amd
Dec 13, 2016 at 7:12

I am aware that there have been already some answers to this question. However, what helped me really understand the concept was the following interpretation with some narrative, as opposed to just a picture.

1. Imagine a horizontal and a vertical line crossing
2. Imagine a unit circle (radius of one) with the centre at that crossing
3. Draw a line at angle θ from the centre to the circle's boundary and beyond
4. Draw another vertical line that touches the circle boundary
5. The distance from the centre of the circle and the point P when the two lines above intercept is the secant of theta This is very well explained at http://www-personal.umich.edu/~copyrght/image/books/Spatial%20Synthesis/trig/ . You can find there an explanation for cosecant as well, among other concepts.

• This does not add anything new to the question as this is already in Michael Hardy's answer. Nov 24, 2021 at 1:57

Sin, cos are circular functions of angle $$(\theta)$$ they are resolved components of a unit circle as is well known

$$sin^2 \theta + \cos^2 \theta =1$$

Using inverse functions definitions as you requested we get

a hyperbola like ( but not a hyperbola) curve plotted on x-, y- axes as shown. The curve does not exist in range/domains $$x=\pm1,y=\pm 1.$$ It can be parameterized for $$\angle POX= \theta$$

$$a=1, x= a \sec \theta, y= a \csc \theta\;;$$

Although shown, it is rarely used in that form in usage. The labeled circular functions are more in use.