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

Question

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.

Answer 1

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.

Answer 2

${{{{{{{{{{{{{{{{{{{{\qquad}}}}}}}}}}}}}}}}}}}}$

Answer 3

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.

Answer 4

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.