Geometric Trig

Trigonometry, circles and triangles combine in this short challenge.
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

Problem

 

Image
Geometric Trig
In this diagram OA is a radius of a unit circle. The hypotenuse of the large triangle is tangent to the circle at A.
 
Find the lengths $\cos(a)$, $\sin(a)$, $\tan(a)$, $\frac{1}{\cos(a)}$, $\frac{1}{\sin(a)}$  and $\frac{1}{\tan(a)}$ in the diagram.
 
Find the areas of all of the regions in the diagram.
 
 
 


Did you know ... ?

Whilst trigonometric functions are defined algebraically in more advanced applications, geometric images such as this one can give great insight into the relationships between the functions. They also impart a sense of the beauty and interconnectedness of mathematics, which inspires many students of mathematics.