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.

Science, Lift, and understanding

A while ago Scientific American published an online article whose title asserted "No One Can Explain Why Planes Stay In The Air". This made quite a stir and there were a lot of responses to it, most repeating or confirming an assertion slightly different than that title. Namely, "Science doesn't understand why planes fly". This is wrong for at least two reasons. First, coming to this expanded conclusion requires a profound misunderstanding of what science is. Second, the problems cited in this article are a misrepresentation of the actual "problem" in the (incorrect) textbook explanation and the gaps that still remain in our understanding of aerodynamics. I'll consider these in order.

First, what does it mean to say the "science" "understands" something? It does NOT mean that there is an explanation that will produce a feeling of intellectual satisfaction in even the most naive person. Some particularly simple things can be explained to this level but not many. There is almost always some amount of background needed to understand what the scientific answer involves.

It would be nice if this weren't the case. Then our intuition would be a reliable guide to understanding the natural world. But it isn't. That's why we have science, that's why we need science. Science works by making testable models of the physical world that accurately predict the outcome of experiments. This is clearly not understood by the writer of the SciAm piece which contains this rather shocking statement " ... by themselves, equations are not explanations ...". This could not be further from the truth.

When a set of equations allow a wide range of phenomena to be accurately predicted we say that science has explained this aspect of nature. There is little in science that our built in intuitions can actually fully embrace. That doesn't mean that science has failed to explain it, it means that we are using the wrong metric to evaluate that explanation.

Second, what are the shortcomings of the standard explanations of flight and how well do the complete models predict the results of experiment. There are two widely proclaimed "explanations" of lift. The Bernoulli explanation and the Newton explanation.

The first is the one in most textbooks. It says that if a small volume of air hits the front of the wing it will split in two and part will go over the wing and part will go under and meet up at the back of the wing. Since the top of the wing is curved that path is longer so the air on top of the wing is moving faster. The Bernoulli effect tells us that faster moving air has a lower pressure so the pressure is higher at the bottom than the top to the wing produces lift. This is almost certainly what you were taught at some time in school.

The most common alternative says that the angle of the wing pushes the air down so by Newton's third law the action of pushing the air down produces a reaction of lifting the wing up.

The Bernoulli explanation has a number of problems. The central assumption that the two parts of the initial volume of air meet up at the back of the wing is simply assumed without explanation. And even worse, that assumption is dead wrong. If you study the airflow over a wing the air going over the top reaches the back of the wing well before the air going under.

The second one is correct in what it states but in this form misrepresents the subject as extremely simple. In reality it isn't quite this bad because every example I've seen includes a statement like: There are lots of other details but this is the dominant effect. The main problem with the approach is that it tells us nothing about the details of the airflow. It doesn't tell us why the air is deflected down.

Neither of these have enough detail to qualify as scientific explanations as I described above since they lack a way to predict an amount of lift that can be compared in detail to experiment. They both succeed at the hand-waving level though. The Bernoulli explanation predicts that a curved airfoil will work better than a flat one and that's often true. The Newton one predicts that lift will increase with a higher tilt to the wing which is also true within some range of conditions.

So what's the real deal? A full treatment is beyond the scope of a blog post and beyond my understanding of the topic. But I can sketch out the ingredients and make enough connections to give what I hope is a cogent picture.

We need to start with a disclaimer. This will not, and can not, be a "complete" description. Air moving past a wing is a collection of an enormous number of individual molecules. The way these molecules act under all circumstances is far too complex to be well and reasonably completely understood because we simply don't have the ability to track all of the details of each molecule of ANY but the very smallest of things. This means that approximations must be used that will result in some details being lost.

The basic assumption is that the air will be considered to be a fluid. Everything I said about air is also true about fluids but we have an intuitive concept of a fluid as a continuous substance NOT made of individual particles. We need only consider the bulk properties to know everything the model can tell us about the fluid and this is enough for almost any situation. In technical terms this problem is solved using an analysis tool called Computational Fluid Dynamics and as grandiose as that sounds the reality is both more complex and the power even greater. For this case we just need to know the density, pressure, and velocity at any point. So don't expect a "complete" explanation. That is unreasonable. But as we increase the number of parameters that we pay attention to we can capture any aspect that is required.

Let's look at some actual cases. I'm using the software from the NASA Glenn Research Center. This software uses takes into account the most important considerations in fluid flow. For simplicity we'll take a look at a symmetrical airfoil parallel to the airflow. The image shows the way air flows past the airfoil as dashes with the speed of the air shown as the length of the dash. Each set of lines shows the motion of a equal amount of air.

What can we tell from this? The dash at the left (leading) edge of the wing is shorter than all of the rest. This tells us the moving slower. The dashes about 2/3 of the way back are the longest and this shows the higher speed at these points. We can also get pressure information from this software but not on the wing graphic. But the Bernoulli Principle tells us what we need to know. Where the speed of the air is low the pressure is high and where velocity is high pressure is low. What is happening here is pretty simple. As the air moves toward the wing it is compressed because it's motion is blocked by the wing. Also the flow lines are closer together near the wing. They have to be because the same number of flow lines are present at each point along the horizontal in the diagram. Since the wing is present the flow lines get bunched. Since same amount of air is the along each flow line to get the same amount of material through a smaller space it must be moving faster.

Here's what happens when that same shape is tilted up by 5 degrees.

Again the highest pressure is at the front and it is lowest about 2/3 of the way back. But you can see that the air is moving much faster across the top of the wing then the bottom. This means that the pressure is lower on the top than the bottom so we have lift.

That sounds a lot like the Bernoulli explanation, if you ignore the fact that the whole motivation given in the textbooks is wrong.  I've presented these two as alternatives because that's the way it is usually portrayed. But that's a big mistake. A better way the look at is to see detailed use fluid dynamics to determine how the flow of is disturbed by the wing. We can either look at the net change in air flow and use or Newton's laws to see what forces are involved OR compute the pressure on the different parts of the wing and get the net force. These are just two different ways to get the same answer and which one is used depends on what we are looking at. If we are considering the details of flow around the wing is makes the most sense to look at it from the Bernoulli point of view. But to figure what that means to the entire plane we need to consider it from the Newton view. On the other hand if we are looking at the motions induced in the air caused by the plane than using the Newton view makes the most sense. Of course to actually compute it you need to consider the Bernoulli effect in all of it's glory.

If you'd like more detail I suggest taking a look here for a more complete treatment.

In conclusion; we DO understand why planes stay in the air. Our understanding is a scientific one that acknowledges the limitations of both ourselves and our tools but they can be made as precise as we need. Perhaps the entire problem could be avoided if the author of that SciAm article had used the title "I Can Not Explain Why Planes Stay In The Air". That would have been correct.

The Most Misunderstood Hubble image

There is a beautiful set of images from the Hubble Space Telescope (HST) that are almost always described as an explosion. Here's a video assembled from those images. It is strikingly beautiful and it certainly LOOKS like an explosion.

But it isn't. The first clue that there is something "funny" going on here is the time scale involved. Let's take a look at the first four individual images used to make that video.