On Cotangents, Tangents, Secants, And Cosecants On Unit Circles.

Question

Above is a diagram of a unit circle. While I understand why the cosine and sine are in the positions they are in the unit circle, I am struggling to understand why the cotangent, tangent, cosecant, and secant, are where they are on a unit circle. Can someone please explain to me?

Answer 1

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{|BE|}{|BC|} = |BE|$

since $|BC|=1$. Similarly

$\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{|AB|}{|BC|} = |AB|$

The placing of the functions in the diagram is purely to do with finding a triangle in which one angle is $\theta$ and the denominator of the relevant ratio is 1 because it is a radius of the circle.

Answer 2

Hint:

$$ \sin{\theta} = \dfrac{sine}{1} = \dfrac{cosine}{cotan} $$

also, $ \tan \theta $ is defined as follows (unit circle):

_{image taken from http://www.rasmus.is/uk/t/F/Su36k02.htm}