# What Sin, Cos, and Tan are, your choice.

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What Sin, Cos, and Tan are, your choice.

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If you draw a circle wir radius 1 around the origin of a coordinate system, and you pick any point on that circle and draw a line to the origin, this line forms an angle with the x-axis.

The x-coordinate of your point is the cos of the angle, the y-coordinate is the sin of the angle. You find a picture of this under the term “unit circle”

A right triangle is a triangle with a single 90ﾟ angle.

The side that is opposite to that angle is known as the hypotenuse.

Each of these terms is a ratio that can be determined from one of the 2 other, non right angles.

The sin is the ratio of the opposite side over the hypotenuse. The cosign is the ratio of the adjacent side over the hypotenuse. The tangent is the ratio of the opposite side over the adjacent side.

So if you have a right triangle whose other angles are 30ﾟ and 60ﾟ, the sin of 30ﾟ will be the length of the smaller side over the length of the hypotenuse.

The reason this is so useful is because it allows you to indirectly determine the length of any of those side’s based on knowledge of angles and the length of other sides.

They’re ways of picking out components of circular motion. Imagine the second hand of a clock. It moves in a circle, but maybe you only want to know how much it’s pointing up or down, or only how much it’s pointing left or right. Sin and cos are used to calculate that from the angle of the hand (the actual angle is measured weirdly offset and backwards for a clock, because clocks weren’t used when they came up with the system). Tangent gives you both at once, by dividing the up-or-down-ness with the right-or-left-ness to give you a ratio.

These functions, and some others, are useful for translating between circular motion, measured in degrees, to motion on a grid, measured by an x and y axis. This can be used for things like drawing circles on a screen, rotating 3D objects in video games, and controlling electric vehicle motors precisely.

All three are known as trigonometric functions. They are the most basic 3 of the trig functions. All 3 have inverses of Sec (secant), CSC (cosecant), and Cot (cotangent). Every single one of them can be defined entirely by any other. For example, Cot(x) = Sin(x+2π)/Sin(x) which isn’t necessarily important to know. However, what is important is that memorizing more than one is not necessary to do all the math related to all 6 functions. It is extremely helpful to memorize the big 3, being Sin (sine), Cos (cosine) and Tan (tangent) because they can make the math much much shorter and simpler.

I tend to be a minimalist. I memorize fewer equations and understand them really well, but I might be a bit slower at getting to the solution because I haven’t memorized the tips and tricks to reorganize things into other forms. Trigonometry has about a metric bajjilionton of proofs and formulas that you can memorize (half-angle formula, double-angle formula, law of sines, law of cosines, etc…) But I made it through an entire physics and math degree only remembering 2 things from trigonometry. The first is the Pythagorean theorem.

The second is the mnemonic SOH CAH TOA. Each of those 3 letter groups represents 3 things. The first letter is the trig functions Sin, Cos, and Tan. If you have a right triangle and know one other angle (besides the right angle giving it the right triangle property), then the sin, cos, or tan of that angle represents a ratio of 2 of the sides which are the other 2 letters. All angles in any triangle touch 2 sides. For the complementary angles in a right triangle (this excludes the right angle), one of those sides is always the hypotenuse (the longest side giving us the H in SOH and CAH). The other side is called adjacent (giving us the A in CAH and TOA). And since there is always a 3rd side in a triangle which does not touch that angle, this is called opposite (giving us the O in SOH and TOA).

Sow how to use the mnemonic, SOH means the Sin(angle)=Opposite/Hypotenuse. CAH means the Cos(angle)=Adjacent/Hypotenuse. And TOA means the Tan(angle)=Opposite/Adjacent. This allows you to fill in all the missing pieces of the puzzle. All you need is one side length and any other piece of information about the triangle and you can fill in every unknown (I’m excluding the right angle from this because that’s a given, so you need 3 pieces of info I guess). And in physics, every single problem can be split into an X, Y, and Z component, solved separately with very simple equations that only deal with 1 dimension, and then combined at the end.

But to recap your actual question, these functions are fed the angle (not the right angle) in a right triangle and they tell you the ratio of the lengths of 2 of the sides. This is only defined from 0 to 90 since those are the min and max allowed values for that angle in a real right triangle. But if you pretend that the direction of the sides matters (like if the triangle sits in the IVth quadrant on a graph, then it can have negative side lengths) well the. We can extend the definition to any input angle. That’s where the unit circle comes in. I’ve seen some other answers covering that already, but ask any questions you’ve got about it and I’d love to throw my hat in the ring and try explaining that too.

Sine is a function. It is a maths thing where you give it an input number and it gives you some output number.

The word sine comes from the Latin sinus, the word for a pocket or lap, from the Arabic “jayb” meaning an opening or fold in a garment (and hence bosom or heart), as a misreading of the Arabic word for a chord (or the sine function itself), from the Sanskrit “jyā” for a bowstring. The basic trig functions have been in use for over a thousand years.

The bowstring thing comes from the fact that a bow, very roughly, forms the shape of two right-angled triangles, and the “sine” function very roughly tells you how long the bow is compared to the bowstring, when you’ve pulled it a certain amount.

At the basic level sine is all about triangles (although later on it turns out to be about circles).

The interesting thing about triangles is that if you scale them up (keeping the angles the same), the lengths of the sides stay in the same ratio; if one is double the length of another, no matter how big you make the triangle, that one will always be double the other. If you change angles, that ratio will change.

So the **sine** function gives us one of these ratios, specifically for a right-angled triangle (where one angle is 90 degrees, or a quarter circle). It tells us proportionally how much smaller the side furthers from the angle we’re looking at is than the the longest side (the one opposite the right angle). This is useful as if we know it for one triangle we know it for all similar triangles.

The “co” in **cosine** stands for “complement” (well, technically complementi, being Latin). Complementary angles are angles that add up to 90 degrees. So in a right-angled triangle, one angle must be 90 degrees, and the other two will be complementary angles (as the total angles inside a triangle must be 180 degrees). So the cosine of an angle is the sine of its complementary angle. It tells you the sine of the other angle in the triangle, or the ratio of the side next to your angle with the longest angle.

The problem with the right-angled triangle definition of these things is that it only works for angles between 0 and 90 degrees. But it turns out an easy way of extending sine and cosine beyond 90 degrees (and below 0 degrees) is to [use a circle](https://www.mathsisfun.com/algebra/trig-interactive-unit-circle.html) (remember that the size of the circle doesn’t matter, as we only care about the ratio of lengths – if we double the circle we double all sides, so the ratio stays the same). In that circle, as we go around it, for a particular angle from the centre, “sine” tells us how far up we are, “cosine” tells us how far along we are, and “tangent” tells us the length of the tangent to the circle at that point (more usefully it tells us the gradient of the line from the centre to that point, but that’s another matter).

One of the interesting things about the sine and cosine functions is that the rate at which each one changes (so how much sine goes up if you change the angle a bit) is proportional to the other one. This leads to them being very useful as solutions to certain maths and physics equations, particularly anything that is cyclical, i.e. repeats in a pattern (such as waves or signals). Sine and cosine then appear all over physics when dealing with waves or repeating patterns, and as quantum mechanics is all about things having wavefunctions, trig becomes very important when dealing with quantum mechanics (although usually we’ll be sneaky and use complex exponentials instead, but that’s another story).