Here’s a pretty thing that got completely missed in my grammar school mathematics class. We build a geometric diagram in which each trigonometric function corresponds to the length of a line.

Remember the trigonometric functions? You’ll need them to appreciate what I’m going to show you. If you’ve forgotten, here is a review. If you remember trig functions (sine, cosine, tangent, secant, cosecant and cotangent) then skip this section.

### Review of trig functions

Any angle θ (Greek letter ‘theta’) between two straight lines generates six useful numbers: the sine ( abbreviated sin(θ) ), the cosine ( cos (θ) ), the tangent ( tan (θ) ), the secant ( sec (θ) ), the cosecant ( csc (θ) ) and the cotangent ( cot (θ) ).

We can calculate these six numbers by making θ the base angle of a right-angled triangle.

In this diagram, θ is the size of angle BAC. Angle BCA is the right angle.

The three most used trig functions are the sine, cosine and tangent.

The other three functions are the reciprocals of the sine, cosine and tangent.

### End of review; on with the good stuff

These ratios are the same no matter how big or small the triangle is. Increase each of the sides by a factor of 10, for example, and

,

same as before.

That last observation means that we can insist that c = 1 and so long as θ remains the same, the trigonometric functions have the same values.

In fact, if c = 1, then

and

So the two sides BC and AC have lengths sin(θ) and cos(θ) respectively.

Now take the triangle with c = 1 and draw a circle around it, a circle with centre at A and radius 1. A circle with radius equal to 1 is called a unit circle.

Now draw the tangent to the circle at B and extend AC to meet the circle at D and the tangent at E.

Since angle ABE is a right angle (a tangent always makes a right angle with the radius), triangle AEB is similar to triangle ABC. Specifically

angle EAB = angle BAC (= θ)

angle ABE = angle ACB (both right angles)

so angle ABE = angle ABC

So, from triangle AEB, we get the following.

Finally, draw a line segment from A parallel to CB that meets the tangent at F.

This creates a triangle AFB with a right angle at B, hypotenuse AF and angle AFB equal to θ. It takes one line of proof to establish that last fact. In this triangle

and

In our complete diagram, therefore, the value of each trigonometric function is represented by a line of that length. Very Greek.

This was completely new to me when I discovered it recently. Do most people with a high school trig course behind them know this?