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 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.
- Imagine a horizontal and a vertical line crossing
- Imagine a unit circle (radius of one) with the centre at that crossing
- Draw a line at angle θ from the centre to the circle's boundary and beyond
- Draw another vertical line that touches the circle boundary
- 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.
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.