What are sin and cos and why are they used so frequently in math?
In: 6
The represent the ratios of length of sides in a triangle given the angle between them. This is why their values are always between -1 and 1.
They also are used in circles and any application where something is rotating. This is why the power in your wall outlets is a sine wave. Back at the plant, a big circular die is rotating within a magnetic field.
Since circles and rotation are everywhere in our lives, so are sin and cos.
Sine and cosign are fundamental concepts in trigonometry – the study of triangles. As it turns out, trigonometry comes up **a lot** in math – so much so there is an entire branch of math dedicated to studying them. They are both expression of the relationship between angles and sides in a right triangle:
– Sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse),
– Cosine is the ratio of the length of the adjacent leg to that of the hypotenuse.
Sin and cos (as well as tan) are very useful functions for working with triangles (specifically ones with a 90 degree angle). They relate angles to a ratio between the sides. So if you know what one of the angles is and the length of a side is, you can find the other dimensions and angle. The cosine (cos) of an angle is the adjacent side (the one touching the angle) divided by the hypotenuse (long side). Sine (sin) is the opposite side over the hypotenuse. Tangent (tan) is the opposite over adjacent.
The real power of this is that it doesn’t have to be a literal triangle, it can be something that you can kinda look at like a triangle as well. So you can apply it to a bunch of stuff pretty easily. Pretty much anything with an angle can utilize these functions. Same with a circle (what is a circle but a line swept through a 360 degree angle). And if a circle can do it, then so can repeating patterns.
As an easy example, let’s say you’re pushing kinda sidewise on a box. Sun and cos will tell you how hard you’re pushing in each direction. Heading northwest? Sin and cos tell you how much north and how much west.
Imagine a circle.
All circles are the same shape, so it doesn’t matter how big, but to ensure we are on the same page, let’s say it is a circle with a radius of 1. “A unit circle ”
Imagine a dot travelling around the edge of the circle. It moves at a steady pace. Do you see it?
Okay, now, you see how that point goes up and down over and over? Let’s track that movement. Ignore the side to side movement, just focus on vertical distance from the centre of the circle.
The dot has to start somewhere, so let’s say it starts off to the far right and we’ll let it move anticlockwise.
So the height will be 0, 1, 0, -1, 0, 1, 0, -1 as it travels around. With many values in between.
This graph is called sine. We shorten it to sin often.
If you track side to side movement inatead you end up with the same pattern (a circle is the symmetric after all), but because we start of the far right, you will start with 1, 0, -1, 0, 1, 0, -1, 0. This is COsine. We often shorten it to cos.
I’d encourage you to Google image search for “unit circle trigonometry gif”. It will show you what I described in motion.
In short: sin and cos are function. You give them a number and they spit out a number.
The number they spit out follows an pattern that’s a wave. The output bounces between -1 and 1.
Why are they used so much? Because the rate that they bop between -1 and 1 maps to how far up or down (for sin), or side to side (for cos) is for a particular angle. This is useful for right angled triangles because because we can take the long edge of a right angle triangle, an angle the edge is pointing and then we can tell how much of that is up and down, and how much is side to side (the other edges of the triangles). This is useful even when we don’t have a physical triangle, like you can take the speed of a rock thrown, the angle it was thrown at and then know how fast it’s moving up and down or left and right, so we could tell where it would hit the ground for example. Any vector ( a thing with a magnitude and direction) can be broken down to get how much of it is moving up and down or side to side .
Additionally theyre used to represent things which go up and down. Like waves. We can change their frequency and amplitude so if we just find the numbers we can then use sin waves to represent anything that goes back and forth regularly, like a swing. We could then use that to predict where that swing would be at any given time.
So they’re useful for triangles imaginary triangles and things which go backwards and forwards a lot. That’s a lot of things. So they pop up a lot.
When you mark a point on a wheel and spin that wheel, the sine tracks the movement of that point in the up-and-down direction (if you plot the sine versus the angle, it makes the characteristic “sine wave” graph). The cosine tracks the movement in the left-right direction, which gives the same “sine wave” graph, just one quarter of a turn ahead.
The reason they are used so frequently is a) that a lot of natural processes tend to produce sine waves, and b) that the rate of change (or “derivative”) of the sine is the cosine, which is the same shape, just shifted by one quarter turn. This gives some neat mathematical properties, which means that mathematicians like to describe all kinds of waves as the sum of different sine waves.
Sine (shortened as sin) and Cosine (shortened as cos) are ratios of different sides of a triangle where one angle is a right angle. The length of one leg divided by the length of the hypotenuse and the length of the other leg divided by the length of the hypotenuse. The references are the angles of the other corner of the triangle. These are important because these ratios are useful as they are uniform regardless of the size of triangles. Even if things don’t look like they are triangles, using these methods we can make all kinds of calculations, especially if circles are concerned, as the ratios are still relevant in pretending their is a triangle inside a circle with the same properties, with the hypotenuse being the radius of the circle.
First you draw the unit circle, although it can be a circle of any size if you adjust things slightly. Next draw a line coming out of the origin and it will automatically hit the unit circle at some coordinate (x,y).
This line you draw will also create an angle with the positive x-axis. Different lines you draw create different angles. Each of these angles is in correspondence with a point (x,y) on the circle. That is, you draw a line, this creates an angle, and the line touches the unit circle at a point.
Due to the correspondence between an angle and a point on the unit circle created by this process we use a short hand notation. When someone writes “sin(angle)” they mean the y coordinate of the point corresponding to the angle. Similarly, “cos(angle)” just means the x coordinate of the point corresponding to the angle.
Once you’ve established this meaning for sine and cosine you can start thinking about them as functions that turn angles into numbers between -1 and 1 because again, an angle you plug in corresponds to an x or y coordinate of a point on the unit circle. Since it’s a unit circle the points only go as far up as 1, right as 1, down as -1, and left as -1.
Now that you have functions you can try to do things like graph them as you would any other function. At this point the input angles become x-values for points on the graph, and the output coordinates between -1 and 1 become y-values on the graph.
These two functions have very nice properties because of how they are defined. For one, they are periodic functions, meaning they repeat the same pattern over and over. They also obey nice identities due to the fact that they turn out to be the side lengths of right triangles.
Do you remember the Pythagorean theorem for right triangles?
> a^2 + b^2 = c^2
Well, sine and cosine are closely connected to that idea, but they relate all of that ratio-of-sides info into something expressed in terms of angle. How?
> a^2 + b^2 = c^2
> (a/c)^2 + (b/c)^2 = 1
> cos(x)^2 + sin(x)^2 = 1
This is helpful because it can help us translate geometry problems involving angles into forms that can be solved by the Pythagorean theorem, or in reverse it can help us identify an angle based on the info we know about the sides of a right triangle.
Imagine a rider on a carousel. You are looking down on it from above (as if in a helicopter). The rider starts due east of the center point of the carousel, and begins to travel around, making an always changing new angle, measured from the current position, the center of the carousel, and the starting “due east” point.
You are keeping track of how far north the rider is from the center of the carousel, as a percentage. The farthest north they can go is 100% north, and the farthest south they can go is “-100% north”. The sine [of the current angle] measures “northness.”
You are also keeping track of how far east the rider is from the center of the carousel, as a percentage. The farthest east they can go is 100% east, and the farthest west they can go is “-100% east.” The cosine [of the current angle] measures “eastness.”
At any point along the ride, you can describe how much north and east the rider is based solely on the angle.
Usually in math, we prefer “1 and -1” instead of “100% and -100%” but they mean the same thing. That’s why sine and cosine values are usually a decimal like “0.375” instead of “37.5%.”
Anything that moves back-and-forth in one or two dimension (like north and east) follows some variation of the sine/cosine rules of “northness” and “eastness” even if the exact formulas describing their motion need to be adjusted.
And this works for other repeatable things, too, even if they don’t physically move. Sine and cosine just describe a position/distance/extent in perpendicular dimensions like north/east or x/y or voltage/time.
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