How To Intuitively Understand Sin And Cos?

The concepts of Sin and Cos appear in a variety of places in mathematics and science. Yet, many folks struggle to understand why we need these concepts.

Take mathematics involving frequencies for instance. How are the notions of Sin and Cos related to frequencies? Well, by the end of this essay, you will laugh about how simple the relationship actually is.

I remember when I first encountered these concepts as part of “trigonometry”. It was a nightmare for me; I think that in most cases, issues start arising when one distances oneself away from first principles.

“Trigonometry” is just a fancy word for the branch of geometry that deals with triangles. To intuitively understand Sin and Cos, we don’t even need trigonometry. All we need is its parent: geometry; more specifically, one particular geometry.

If you think about it, everyone has an innate understanding of geometry. All folks can grasp what a circle is, regardless of their technical abilities. A circle is all that we will need to intuitively understand Cos and Sin. Let us begin.

This essay is supported by Generatebg

The Magic of the Unit Circle

Now, for our purposes here, we could work with a circle of any size. But for simplicity and scalability, it makes sense to work with a unit circle.

This is nothing but a circle whose radius is 1 unit long. Next, let us place a simple 2-dimensional Cartesian coordinate system at the centre of this circle as follows:

As you can see, the origin of the coordinate system matches with the centre of the circle. Up next, let us imagine a point moving along the circle in the anti-clockwise direction as follows:

With this setup, we are ready to intuitively understand how Cos and Sin work. Let us keep going.

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Understanding Cos

As the point moves along the circle in the anti-clockwise direction, for the sake of simplicity, let us track its position at four significant locations where the circle meets the x-axis or the y-axis.

The coordinates of these intersection points are as follows:

Let us now say that the point starts on the right-hand side. For now, we are only focused on the x-coordinates of this point (you’ll understand why in a bit). It starts at x = 1 (0 radians along the circle).

When the point is at the top of the circle (π/2 radians along the circle), the x-coordinate equals 0. When it is at the left most point (π radians along the circle), the x-coordinate equals -1.

Finally, when the point reaches the bottom-most point of the circle (3π/2 radians along the circle), the x-coordinate equals 0.

These x-coordinate values are nothing but the Cos or Cosine values with respect to this circle. In other words, Cos is a function that tracks only the x-coordinate values of a point moving around a unit circle.

To summarise:

1. Right-most point → Cos(0) = 1

2. Top-most point → Cos(π/2) = 0

3. Left-most point → Cos(π) = -1

4. Bottom-most point → Cos(3π/2) = 0

Next, let us move on to Sin; you might already have an idea about it at this point.

Understanding Sin

This time, as the point moves along the circle starting from the right-most intersection point, let us focus on the y-coordinate values only.

At the starting point (0 radians along the circle), the y-coordinate equals 0. At the top-most point (π/2 radians along the circle), the y-coordinate equals 1.

At the left-most point (π radians along the circle), the y-coordinate equals 0. And finally, at the bottom-most point (3π/2 radians along the circle), the y-coordinate equals -1.

In other words, Sin or Sine is a function that tracks only the y-coordinate values of a point that moves along a unit circle.

To summarise:

1. Right-most point → Sin(0) = 0

2. Top-most point → Sin(π/2) = 1

3. Left-most point → Sin(π) = 0

4. Bottom-most point → Sin(3π/2) = -1

When I first encountered these values in school, I was told to memorise them. It does not have to be that way. As you can see, with first principles, we need not memorise unnecessary information.

All this is great! But how are Cos and Sin related to frequencies?

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How to Intuitively Understand Sin and Cos?

We use frequencies in the context of cyclical phenomena. Anything that operates in cycles is metaphorically going in “circles”. Aha!

So far, we have been imagining a point moving along a unit circle. Now, imagine that this point is moving along the circle at some rate.

If the point completes one revolution in one second, we say the frequency is 1 Hertz (Hz). Similarly, if the point completes 2 revolutions in one second, the frequency is 2 (Hz). And so on.

This framework finds many practical applications in fields such as signal processing, audio analysis, and wave propagation.

The concepts of Cos and Sin allow us to represent and manipulate cyclical phenomena using a mathematical framework.

Why is this important? Well, it saves us a lot of time and cost, as we can simply simulate phenomena instead of actually replicating the physical behaviour each time around.

I hope you found this essay useful. The next time you come across Cos or Sin in the context of frequencies, I hope you remember the geometrical point moving along the unit circle!

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Further reading that might interest you: 

• Why Do We Really Use Euler’s Number For Growth?
• Logarithms: The Long Forgotten Story Of Scientific Progress
• Calculus: How To Really Understand A Function

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