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I've asked a question on Math Stack Exchange, but it was suggested it might be a better idea to post it on this Educators instead.

Here's the question link: https://math.stackexchange.com/questions/3545998/intuition-behind-sin-theta-when-introducing-this-to-high-school-students

Briefly, the question was:

Suppose I'm introducing trig, and I ask students to imagine trying to figure out the height of a tree, if you know how far you are from the base of the tree and you estimate the angle to the top of the tree.

If you take the tree example from above, we have the adjacent side and the angle. Now:

The definition of $\tan(\theta)$ is the missing quantity we wanted in the first place. The ratio of the opposite side and the adjacent side. But how does $\tan$ go and calculate the ratio when I give it an angle? Is it just some magic computer?

I think it's possible to convince them - once I have this ratio, I can find the length of the missing side: $\text{Opposite} = \tan(\theta)\times \text{Adjacent}$.

Clarification: Thanks to those in the comments for responding and suggesting clarifications.

  • Yes, how can we explain to you how a calculator computes, say, tan($62^∘$)? And I'm also looking for ideas for how to then explain that to students?
  • A method on how it might be computed, which is a little more general then asking them to imagine a infinitely large table of values where we've taking heaps of right angled triangles and calculated their ratios
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    $\begingroup$ I'm a little confused about what you're wondering about. Do you want someone to explain to you how a calculator computes, say, $\tan(62^\circ)$? And/or are you looking for ideas for how to then explain that to students? Or, differently, are you wondering about the cosmic existence of tangent as a function?! $\endgroup$ – Brendan W. Sullivan Feb 14 at 5:31
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    $\begingroup$ Students will have had a fair amount of experience working "applied trig type problems" in the case where the triangles are 45-45-90 and/or 30-60-90, and they are familiar with ("have been exposed to" is probably more accurate) the idea of similar triangles. Thus, I usually started off by explaining that, just like for 45-45-90 and 30-60-90 triangles, there are "rules" for working with 20-70-90, 37-53-90, 10-80-90, etc. triangles. However, instead of remembering something like $\frac{\sqrt{3}}{2},$ you look up the relevant values in a table at the back of the book (later, with a calculator). $\endgroup$ – Dave L Renfro Feb 14 at 7:35
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    $\begingroup$ Regarding how trigonometric values can be calculated (in the sense of obtaining decimal expression approximations), you can often find discussions of this in old trigonometry texts. For example, see Chapter IX: Construction of Trigonometric Tables (pp. 82-93) in Isaac Todhunter’s Plane Trigonometry (1882). archive.org version (1890) $\endgroup$ – Dave L Renfro Feb 14 at 7:43
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    $\begingroup$ Please edit the question to clarify what you're asking. Do you literally want to know how a particular calculator does it? Most likely with Chebyshev polynomials, but that probably isn't documented for any given calculator, and doesn't seem like something that can be taught at this level. Do you instead want to teach them some possible method for calculating something like tan 62∘ from scratch? If so, then what are your requirements for this method? $\endgroup$ – Ben Crowell Feb 14 at 8:28
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    $\begingroup$ Have you read about the CORDIC algorithm? It’s a pretty standard type of algorithm for computing these values, but it may not be simple enough that you would want to try to explain it to students. It involves using a rotation matrix and iteratively approximating sine and cosine. However, one already has to have certain values of arctangent computed and stored. $\endgroup$ – Nick C Feb 14 at 13:29
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If you are teaching this at an introductory level, then the algorithm that calculators use today is going to go far over their heads. (It might go over MY head!) The story of how we developed increasingly accurate trig tables over the course of history would be an interesting topic of inquiry for advanced algebra / pre-calculus / calculus, but at the beginning you should focus on the key ideas for the beginning.

A constructivist introduction to trig would be to put aside the scientific calculator for a while. So you estimate the height of the tree by drawing a right triangle with the same angle of elevation, noticing that the triangle is similar to the real-world triangle by AA, and using a proportion to solve for the unknown height. Then you might extrapolate from that. What if your job was estimating the height of 50 trees a day? Drawing and measuring all of those triangles would take up a lot of your day, especially when you would find yourself redrawing the same triangle when you had the same angle of elevation. So realizing that the only needed input value was the angle of elevation and the only needed output value was the ratio of the two sides, you can replace all of that triangle drawing with a simple table of angle to ratio. So that is a brand new function that we just created to address a real-world problem!

If you feel like sticking on with the constructivist perspective, you can then move on to having the class construct a rudimentary class-wide trig table by having different teams measure the three principal ratios for triangles ranging from $5-85-90$ to $45-45-90$ in five degree increments. You can attach names to the functions at this point and start to develop observations that are important. For instance, it is no accident that the sine of an angle is the same as the cosine of the complementary angle. You can then have the class start solving real-world trig problems, except still using their own constructed trig tables and a four-function calculator.

From here, it's just a simple step forward to say that this is such an obviously important trio of functions that people throughout history have struggled to measure them accurately. Because of engineering, navigation, and ballistics, it's not at all an overstatement to claim that the balance of global power has shifted based on having a more modern trig table than your rivals! Ancient folks like Hipparchus and Ptolemy used geometry (see below). The development of calculus made Taylor series a reality that Charles Babbage and Ada Lovelace used to develop to precursors to the first mechanical computers. And modern computers and calculators can essentially compute the trig values of very precisely measured angles to any necessary degree of precision. But I think the key idea is that we can use the calculator in the same way that students used the table they developed and have a very reliable measurement that they don't have to treat as "magic".


That said, I have long thought that it would be an interesting honors project for a sufficiently motivated precalculus student to try and create a portion of the sine table to a certain decimal precision without technology in a process inspired by Ptolemy's calculations (which may have been Hipparchus' method as well, although Hipparchus' work is lost to history because of the jerks who destroyed the library at Alexandria). Ptolemy's function measured the length of a chord subtended by a central angle of a given size. That function $\operatorname {crd}$ is not in use today, but it is the same as $\sin$ except for scaling.

The key ideas are noting that $\sin 30^\circ=0.5$ and $\sin 36^\circ=\frac{\sqrt{10-2\sqrt5}}4$ (which can be shown with a little trig and factoring a particular cubic polynomial). Then you basically set up the sine subtraction and half angle formulas in terms of sine alone. Doing this and learning the old-school strategy for calculating square roots, you can calculate $\sin 6^\circ$, $\sin 3^\circ$, $\sin 1.5^\circ$, and $\sin 0.75^\circ$. Finally, interpolate the final two values to estimate $\sin 1^\circ$, and then you can combine them with other known values to estimate the entire sine table to the nearest degree.

That's a lot, but it obviously within the realm of achievement. It would be interesting to see how far a student or team of students could get even if they used a spreadsheet to do the work arithmetically.

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The common way to introduce how trig functions are calculated numerically is via Taylor/Maclaurin series. These are used extensively in a lot of engineering and physics based applications.

However, this requires a knowledge of calculus, and isn't very useful in the context of basic trigonometry. It also isn't useful for giving students a "feel" of how these functions work, or how they might have developed them themselves.

Instead, let me introduce you to a more elegant tool from a more civilised time: The Unit Circle.

A unit circle.

(This particular version is my own presentation, created using the excellent pynomo package. Anyone is free to use or adapt it as they wish.)

This is just a very simple diagram consisting of a circle with a radius of 1 arbitrary unit. It has a horizontal and vertical axis, and it has an angular scale marked round the outside in revolutions, degrees and radians.

Introduce it to students by showing them a triangle with the hypotenuse deliberately chosen to be one unit long. Assuming that they have been taught:

$\sin(\theta) = \frac{Opposite}{Hypotenuse}$

and

$\cos(\theta) = \frac{Adjacent}{Hypotenuse}$

Then it should be easy to see for a triangle with a hypotenuse of 1:

$\sin(\theta) = \frac{Opposite}{1} = Opposite$

and

$\cos(\theta) = \frac{Adjacent}{1} = Adjacent$

This means that for the special case where a right angled triangle has a hypotenuse of 1, the horizontal line will have a length equal to the cosine of the angle and the vertical line will have a length equal to the sine of the angle.

A circle of radius 1 represents all points that are 1 unit away from the origin. So if we draw a circle, and then draw a radial line, then the horizontal coordinate of where it meets the circumference will be the cosine of the angle, and the vertical coordinate will be the sine.

You can demonstrate this with the unit circle as follows. Suppose you want to find the sine and cosine of 20 degrees ($\pi/9$ radians). Lay a ruler on the unit circle such that it passes through the centre of the circle, and the 20 degree mark:

Unit circle demonstration

Then, simply read off the horizontal and vertical coordinates where the ruler passes through the circle. you will find that the vertical axis has a coordinate of about 0.34, and the horizontal coordinate of 0.94. If you want to check on a calculator, you will find:

$\cos(20 ^{\circ}) \approx 0.9397$

$\sin(20 ^{\circ}) \approx 0.3420$

If you want to find the tangent of an angle, you can either employ the trig identity:

$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $

or you can place a 2nd ruler tangent to the point where the first ruler crosses the unit circle, and measure the number of units between the tangent point and the projection of the horizontal axis:

Trig construction.

This also helps introduce why it's called the "tangent".

I like using this approach because it is such a simple construction where everything is apparent. There is no "magic" in it, and you can measure any sine and cosine very quickly. You can also have your students play about with it in order to get a feel of how these functions behave. It allows you to introduce inverse functions by asking them what angle is associated with a given $\sin(\theta)$. You can also introduce a bit of maths history about why $\sin$, $\cos$ and $\tan$ are referred to as circular functions.

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There are a lot of approaches to approximating sine and cosine which would be sensible to a high school audience.

One method is to use the double angle formula repeatedly, together with small angle approximations. For instance, if you want to approximate $\sin(\theta)$ and $\cos(\theta)$, approximate $\sin(\theta/64) \approx \theta/64$ and $\cos(\theta/64) \approx 1$. Then use the double angle formulas repeatedly until you have an approximation for $\sin(\theta)$ and $\cos(\theta)$. If you need more precision, use a larger power of two in the denominator (but be prepared to pay with more time calculating). In principle you could do all of these calculations by hand: you only need the 4 basic operations.

I think (but I am not sure) that this is the original reason people cared about the angle sum formulas.

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Like the other answers, I'd also suggest lots of computations for many triangles of increasing steepness. This way students could gradually figure out what the curves are like, what the domains and ranges are, and where it increases, decreases, etc. They could sketch approximations of the functions?

So, for example with $\tan(62)$, even if they don't know what exactly it is, they could at least tell if it is greater or less than similar values like $\tan(60)$ or $\tan(65)$

For degenerate triangles or triangles in other quadrants, I'd start off with a unit circle, note how ratios stay the same for similar triangles, and how $x$ and $y$ values are related to side lengths. This way they can work with "$0$" side lengths as well as "negative" ones. (Relating it to the sketches, they'd be able to know in quadrants the ratios are positive or negative)

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You have to observe that $$ Opposite = \frac{Opposite}{Adjacent}*Adjacent = \frac{Opposite}{Hypothenuse}*\frac{Hypothenuse}{Adjacent}*Adjacent = \frac{\sin(\theta)}{\cos(\theta)} *Adjacent= \tan(\theta)*Adjacent$$.

So you start with an unsual math-trick and multiply by one.

Now you can concentrate on $\frac{Opposite}{Adjacent} =\frac{\sin(\theta)}{\cos(\theta)}$. So you have to understand that when scaling an triangle, the angles and the ratio of the sides stays the same.

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