Trig Functions: A Better Introduction

One of the things I really don't understand is the terrible way that trig functions are introduced. The definitions given are all ratios of the sides of a right triangle. This does give the correct answers but it completely masks the meaning behind the words and obscures some simple ways to understand what is going on.

First, the word Trigonometry means triangle measurement. The value of all of the trig functions can be determined with a ruler by direct measurement. To see what I mean by this let's start very differently from the way you were probably taught: the Tangent function.

First we draw a circle with a radius of one unit, the horizontal axis, and a vertical line tangent at the right edge, like this:

Now draw a line from the center at the angle of interest relative to the horizontal axis out past the tangent line, like this:

The tangent of this angle is the length along the *tangent* line between the horizontal axis and the radial line, the red part shown here:

This is why that value is called the tangent. No dividing or other computation need to be done. Just measure the length.

The word "secant" comes from the Latin word for cut. The length from the center along the radial line that "cuts" through the diagram till the vertical line is the secant of this angle. It's shown in green here:

If we look at the arc of the inside of the circle between the radius and the horizontal line get the last of the basic trig functions. The Latin word for bay is "sinus". The sine of that angle is half the chord of the circle from the intersection of the line and the circle. This diagram makes that clear. The sine of the angle is shown in blue, here:

These are the three basic trig functions: Sine, Secant, and Tangent.  Here are all three of them on one diagram.

The other three are based on the idea of a complementary angle. The complementary angle is the angle that when added to the original angle is 90°. This is the angle between the slanted line and the vertical. So if we add a tangent line at the top of the circle and a y-axis the we can find the corresponding lengths for these three complementary functions. The sine of the complementary angle is defined as the cosine. The other functions, cosecant and cotangent are obvious at this point. I'll use dashed lines of the same color. The dashed blue line is the cosine, the cosecant is the green plus the dashed green, and the red one the cotangent.

These can be shown to be equal to the textbook definitions pretty easily. To make this match the textbooks I'll switch to the standard 3 letter forms:
sin=sine
sec=secant
tan=tangent
cos=cosine
csc=cosecant
cot=cotangent

We can see that sin/cos=tan by looking at these two similar triangles.

The smaller triangle is clearly similar to the larger one because all three angles are the same. Note that the base of the smaller triangle is the same length as the dashed blue line. For the larger one the ratio of the sides that aren't the hypotenuse is the length of the red line to the radius of the circle which is 1: tan/1=tan. For the smaller one that ratio is sin/cos so we get that sin/cos=tan.

Looking at the same two triangles we see that the ratio of the hypotenuse to the horizontal side of the smaller one is 1/cos. For the larger one it is sec/1 so we get the 1/cos=sec

Looking at the next two triangles we see that the ratio of the hypotenuse to the vertical side of the smaller one is 1/sin. For the larger one it is csc/1 so we get the 1/sin=csc

The most cited trig identity sin²+cos² = 1. This can be read off the diagram by looking at this triangle and using the Pythagorean equation. The square of the hypotenuse (which is just 1² or 1) is equal to the sum of the squares of the other two sides.

Similarly this shows that 1+tan²=sec²

And 1+cot²=csc² is shown here:

I think that this is a far better way to introduce the trig functions. It motivates the names, allows simple constructions for the simple relationships between them, and is easier to remember.