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Geometric Trig

Age 16 to 18 Short Challenge Level:

 

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.