I read an article today on the six basic trigonometric functions, and I thought there was a particularly important insight that I wanted to present in my own words.
When I was in school, we learned the six basic trigonometric functions. Since I’ve been teaching, I’ve noticed that only three of these are emphasized: Sine, cosine, and tangent. The other three are the inverses (cosecant, secant, and cotangent, respectively).
However, notice that three function names have co- in front of three other functions names, suggesting a relationship. More importantly, notice that the three functions lacking the “co-” prefix are not the three that are usually emphasized.
Consider a basic right triangle:
There are three sides. If we set any side to a length of one unit (whatever that unit is), we can then describe the other two sides in terms of their ratio to that side.
For instance, let’s say we set the hypotenuse to one. The red line is going to be shorter than the blue line, and we can determine its length if we know how big the angle \(\alpha\) is. That is to say, \(\frac{red}{hypotenuse}\) will always be the same value, if \(\alpha\) is the same. We call this value the sine of \(\alpha\).
Likewise, the green line is going to have a consistent ratio with the hypotenuse. We call this value the cosine of \(\alpha\).
If we just limited the basic functions to sine and cosine, we could just stop here. The sine and cosine are the values of the two legs of a right triangle with a particular angle when the hypotenuse is set to one unit. Indeed, the tangent is sometimes (but not often, in my experience) primarily defined as the ratio between the sine and the cosine.
However, in order to see the tangent properly, we need to look at its historic definition. Geometry classes generally get to triangles before circles, so by the time students get to tangents and secants of circles, they already have the first term in their minds as being related to triangles.
This is putting the cart before the horse, though, particularly for students who go on to calculus, where tangent and secant lines become very important.
With regards to circles, a tangent is a line that touches a circle exactly once; a tangent always forms a right angle with a radius of the circle. Any line which touches the circle in two places is a secant line. Let’s draw a circle with a secant line through the center of the circle, a tangent line, and the radius that’s perpendicular to the tangent:
Notice that this creates a right triangle. Let’s say this is the unit circle; that is, the radius is one. The red line will have a predictable ratio to the green line, and it’s part of a tangent line: This is why it’s called the tangent. The blue line also has a predictable ratio to the green line, and it’s part of the secant line that also includes a diameter of the circle: This is why it’s called the secant.
That explains the names, but let’s return to the first diagram:
If we set the green line to a length of one unit, then the red line will be the tangent of \(\alpha\) and the blue line will be the secant of \(\alpha\).
So far, we’ve set the hypotenuse and the leg adjacent to \(\alpha\) to one unit in length and named to lengths of the other two sides. We have a third side: The leg opposite to \(\alpha\). Just as we named the side paired with the sine the cosine, we’re going to name these sides the cotangent and the cosecant.
Specifically, if we set the red line to a length of one unit, then the green line with be the cotangent of \(\alpha\) and the blue line will be the cosecant of \(\alpha\).
Let’s lay the names out in the form of a table:
| ↓ Unit Side ↓ | Hypotenuse | Adjacent leg | Opposite leg |
| Hypotenuse | Unit | Cosine | Sine |
| Adjacent leg | Secant | Unit | Tangent |
| Opposite leg | Cosecant | Cotangent | Unit |
We can then use this table to read off each function without setting any sides to units. Remember that these functions represent ratios between sides: The column heads represent the numerator of that ratio, while the row heads represent the denominators. That is:
| → ÷ ↓ | Hypotenuse | Adjacent leg | Opposite leg |
| Hypotenuse | 1 | Cosine | Sine |
| Adjacent leg | Secant | 1 | Tangent |
| Opposite leg | Cosecant | Cotangent | 1 |
For instance, the sine is the opposite leg divided by the hypotenuse, while the secant is the hypotenuse divided by the adjacent leg.
Learning the relations in this way rather than in terms of the mnemonic SOH CAH TOA leads to a deeper understanding of the material.