Brief recap
In goniometry there are three important functions which are known with the name of sine cosine and tangent.
When we studied trigonometry it was explained to us how those functions are related in a unit circle(a circumference in the centre of the xoy diagram with radius 1).
The cosine is the ratio of the adjacent side and the hypotenuse, meanwhile the sine is the ratio of the opposite side and the hypotenuse, and the tangent is the ratio of the opposite side and the adjacent side. Since the hypotenuse in that triangle of the unit circle is 1, the cosine is the x value and the sine is the y value of the point in the unit circle.
And the tangent? The tangent, in the graph, is represented by the line tangent to the point of the unit circle and intersects the x axis.
The reciprical functions' history
In the past there was the necessity to introduce the inverse functions and to give them a name(secant, cosecant, cotangent). That necessity came from the fact that people working with math, in the past, didn’t have a calculator and they were using big tables where it was written all of those values. The usage of those names for the inverse functions of trigonometry came to us and now we are still using them.
Reciprical functions in the graph
So, where are those reciprical functions in the graph? Well, the cotangent is simply the line tangent with the unit circle that goes from the point on the unit circle to the y axis. Now, from the graph we can deduce that the angle theta is congruent to the angle formed by the tangent line and the y axis. For the cosecant, which is 1 over sin theta,we can work as we did for the sine function and so we take the ratio of the opposite side and hypotenuse of the new angle and, since the opposite side is 1, we find out that the cosecant is the hypotenuse of the new triangle which is colored in light blue. In the same way as we did before, we work on the other triangle and we take the ratio of the adjacent side and the hypotenuse and, since the adjacent side is 1, we find out that the secant is the side colored in green.
Kind of cool! don’t you think?
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