Intuition behind $\sin(\theta)$ when introducing this to high school students

Question

When first introducing trigonometry to students, the traditional setup is to start with a right-angled triangle with reference angle $\theta$ and we label the sides with "Hypotenuse, Opposite and Adjacent."

To keep students engaged with some practicality behind this, we can give an example of trying to figure out the height of a tree, know how far you are from the base of the tree and estimating the angle to the top of the tree.

Then we define something arbitrary called "$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$".

I feel like at this point, students lose the conceptual intuition behind what's going on. Some students who are able to just accept it without questioning it too much can start punching in numbers and angles into the calculator when doing example questions. Other students who feel stuck with this weird idea might not be able to move forward.

What would be a good idea to explain how to think about $\sin(\theta) $? I don't want to introduce a unit circle type definition because I feel like it will only make it less tangible for them. Can we do better than something like "it's a magic computer which tells you the ratio of the opposite and hypotenuse sides of a right-angled triangle when you supply it the reference angle"

To maybe elaborate/clarify: I feel like a few things that students might not be able to understand

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 a angle?

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

Answer 1

My answer is more pedagogical than mathematical but the question is asked here, so here goes:
This is an approach I have used with students who are learning about trig ratios for the first time.

First, I make sure that students understand the idea of similarity, similar triangles, and how the ratios of any pair of corresponding sides are equal for all triangles that are similar to each other.

If that's all good, I put to the students that this means that for any given shaped triangle, if we know the ratio of a pair of sides, and we know length of one of these sides in a triangle of that shape, we can work out the length of the other side (discuss with examples)

After explaining that we are going to look at right-triangles (and we talk about why they might be chosen above all others), I issue a worksheet.

This worksheet has a lightly drawn first quadrant with 10 cm radius. Every 5 degrees around the circumference there is a dot. Students are then assigned a particular angle (plenty of double ups to allow for error checking), and are asked to draw a right-triangle incorporating this angle. They are then to accurately measure the opposite side and the hypotenuse, and give me the ratio O/H. I write these on the board building up a table.

Ok, we now have a table from 5 to 85 degrees for right-triangles and can now do some calculations (heights of trees or whatever).

Lastly, I put to students the question, what if we could produce a table for every possible angle? And then hand around a photocopy of a page from my old four-figure mathematical tables book (for Sine) - ok, it's not every possible angle, but.... We use that for a few more examples.

And finally, we get to the calculator. At this point I'm not fussed if students imagine that somehow these tables are programmed into the calculator after someone somewhere has spent meticulous hours of measuring and calculating. The important thing is that they realise (I hope) that these numbers are not just plucked out of thin air, but that there is a solid basis to them.

Answer 2

You can sell sine and cosine based on expressing how much of the right triangle in question aligns with the adjacent or opposite side.

Let us set notation,

• $A$ = adjacent side length
• $B$ = opposite side length
• $C$ = hypotenuse side length

Since the triangle is assumed to be a right triangle we know $A^2+B^2=C^2$. Let $\theta$ be the angle between $A$ and $C$.

• the hypotenuse is the longest side; $A,B \leq C$
• the only way for $A=C$ is that $\theta = 0^o$ (this happens when $B=0$
• if we imagine $A$ shrinking to zero we see $\theta$ gets close to $90^o$

We can introduce sine and cosine as devices to express how much of $C$ is used in forging $A$ or $B$:

• $A = C \cos \theta$
• $B = C \sin \theta$

Notice since $A,B \leq C$ we must have $\cos \theta, \sin \theta \leq 1$. Also, when $\theta = 0$ we noted $A=C$ hence $\cos 0 = 1$ whereas $\sin 0 = 0$. Conversely, from the case of $A \rightarrow 0$ we saw $B=C$ and $\theta = 90^o$ hence $\cos 90^o = 0$ whereas $\sin 90^o = 1$.

Of course, there are much better ways. But perhaps this is sort of in the direction you seek ?

Answer 3

The next section is an introduction/motivation to trigonometry. The presentation does not require the definition of the $\text{sin}$ function - it is an overview.

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Why not just start at the beginning to find how intuition dovetails with the ancients development of trigonometry?

Here is something for everyone to marvel at (forget proofs and embrace its properties with your mind) - the isosceles triangle:

In a natural way every isosceles triangle can be placed inside a circle with its base (an unambiguous notion except when the triangle is also an equilateral triangle) a chord of the circle:

The students should understand that if we know $r$ and the angle $\theta$ that there can be only one corresponding length $s$ for the chord and that it has the form

$\tag 1 s = r * \text{crd}(\theta)$

From wikipedia,

Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function.

Recall that a regular polygon may be characterized by the number of its sides and by its circumradius (or just radius), that is to say, the constant distance between its centre and each of its vertices.

Exercise: Show that the perimeter of a regular polygon with $n$ sides and circumradius $r$ is given by the formula

$\tag 2 P = n * r * \text{crd}(\frac{360°}{n})$

It was wonderful to have such a formula but to be useful the ancients needed a table of chords and some estimation techniques. The students should be encouraged to skim over

$\quad$ Ptolemy's table of chords

to get an appreciation of the powerful calculating devices they get to use in our high-tech age.

Now explain to your students that if one continues working on these type of geometric problems, they'll eventually drop the $\text{crd}(\theta)$ function and prefer working with $\text{sin}(\theta)$, the half-chord function.

For example, we have these formulas for our isosceles triangle:

$\tag 3 \displaystyle r = \frac{s}{2 sin(\frac{\theta}{2})}$

$\tag 4 \displaystyle r = \frac{h}{cos(\frac{\theta}{2})}$

Encourage your students to skim over the closely related article

$\quad$ Radius of a regular polygon

Yes, it is uncomfortable leaving the isosceles triangle behind, but if one pursues their math studies they might reach the point that they see how the modern 'core theory' of trigonometry using $\text{sin}(\theta)$, $\text{cos}(\theta)$ and $\text{tan}(\theta)$ is much more than just 'extra stuff and things' to memorize. Indeed, it allows us to 'come full circle' and work with something called Euler's formula, allowing us to marvel again at

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OK, enough of the overview. Time to get down to business with SohCahToa and carefully examining $\text{(1)} - \text{(5)}$ as a first lesson in this modern trigonometry class.