Geometric Trig
Trigonometry, circles and triangles combine in this short challenge.
Age
16 to 18
Challenge level
Problem
Image
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.
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.
Getting Started
This problem uses basic notions of trigonometry and areas of triangles which are fundamental to developments in A-level mathematics.
Being able to work with diagrams such as this is useful prepation for both coordinate geometry and mechanics.
Student Solutions
There are many ways to create the solution as it is something like a jigsaw. We used this method
1. Mark the right angles (use the fact the a radius and tangent at a point are at right angles)
2. Make all occurrences of the angle a
3. Mark the unit length
The diagram then becomes
Image
To work out all of the areas we need to decide what unit of measurement the angle $a$ is in. Of course, we choose radians: there are $2\pi$ radians in a circle. We will also need to know the formula for the area of a circle and the area of a triangle. In this case, the areas are given as
Image
The two largest areas are equal for around 0.40523 radians (23.2 degrees).