Can someone tell my what cos, sin and tan actually measure?
In: Mathematics
Draw a circle at the origin of a 2D graph with a radius of 1. Draw a line (technically a ray) from the origin. The angle is measured from the positive x axis to the line. (Cosine, Sine) are the coordinates where it hits the circle. Tangent is the slope.
The ratio between the sides of a right triangle.
Once you fix one of the angles of a right triangle (I mean, one of the angles other than the right angle), you fix the shape of the triangle. You can make it bigger or smaller, but the shape has been fixed and so too have the *relative* lengths of the sides. That means if you have one angle and one side you can completely describe the right triangle.
Those trig functions take one angle as an input and output the ratio between two of the sides.
That’s where the functions originally came from, but it turns out that they also show up in all sorts of places that have nothing (obvious) to do with right triangles.
They meassure the X and Y when you moving a dial on a circle. Think of clock where the minute hand moves around: if you draw a line from where the minute hand points strairth down and straight left to project to axis, and where it hit the X and Y axis it meassure that length of the projected line.
Tan is just a ration of sin over cos, and it proves useful for a lot of triginomitry calculations.
let’s say your arm is exactly one meter long, and you’re in a swimming pool, with the surface of the water at shoulder level
if you stretch your arm forward along the surface of the water, and lift it up by an angle A, then
* sin(A) is how far **above** the water the tip of your fingers are
* cos(A) is how far **in front** of you the tip of your fingers are
if you then do the same thing, but this time you’re firmly grabbing a laser pointer pointing downwards in your fist, then
* tan(A) is how far the laser beam has to travel before hitting the surface of the water
Note that when cos(A) is behind you, or sin(A) is underwater, or tan(A) is pointing away from the surface, they give you a negative distance c:
If you have a transparent wheel spinning round with a mark on the rim, sin is how the mark moves up and down, cos is how it moves side to side and tan is their ratio.
Edit: my first animation is not good, this is much better, thanks to armaddon below
https://www.reddit.com/media?url=https%3A%2F%2Fi.redd.it%2Fpouzspkfr6q61.gif
There are already so many great answers here, I only want to add:
[this animation ](https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and_cosine_animation.gif) will _not_ easily explain the concepts. However, if you really study it, it _can_ explain the concepts in a way that words can not.
I would _not_ recommend using this to explain the concepts to someone. It doesnt explain what sin and cos actually _are_, but once you understand what they are, this animation can help you understand how we go from “measuring right triangles” to “describing wave forms”, which is an important step in understanding one of the most common and fundamental models we use to describe the world through a lens of scientific inquiry.
That is to say, if you wanna know what’s physically true, to differentiate fact from fiction, you’ll want to understand waves, and if you want to understand waves, you’ll want to understand trig functions.
[this animation](https://www.reddit.com/r/educationalgifs/s/PykTXjj28r) is much better, but it’s also very busy and includes further concepts like secant and cotangent that you don’t need right now.
They are a ratio. But much more than that. Tan is “less useful” than the others but still useful in some applications, it’s just that tan is directly related to sin and cos. Sine (sin) is the base function here, everything else can be made in terms of it. If you are familiar with the basic definitions, sohcahtoa, then that doesn’t really tell you what they are just how to calculate them for triangles. Even in this, however, you can see the ratios at play, it’s opposite over hypotenuse. That’s a ratio. What they actually are though, is they are a way of measuring phase angles. This gives them a lot of power but they aren’t really a thing, they are a way of converting between types of units, angles to numbers, lengths to angles to numbers, etc. and the biggest power is that measurement of phase angles, allowing you to convert from a wave function to something else, or a circle to a line.
These functions measure ratios of sides of a right angled triangle. But to be more intuitive:
Stand a distance away from the room of your wall and raise your arm at any angle. Now you can probably guess if there was a line from your arm, it would probably hit that wall. The fun thing is, the height at which the line hits the wall has ratios with your distance from the wall and the distance from you to the point where the line touches the wall. Makes sense. Now the real fun part, for a given angle, say, hm, 45 degrees, whether it’s your arm or your friend’s arm, at any distance, the ratio will be the same. In fact, the ratio is same for any right angled triangle at 45 degrees.
Sin, cos, tan, are functions that give us the ratios. Say you know you’re 50 feet away from a 50 feet tall building. Then you know that the angle your feet make with the top of the building is 45 degrees, because the tan function tells us that at 45 degrees, the perpendicular (height of building) is equal to the base (distance from the building). So it’s real helpful in calculating stuff, cus everything likes to be in nice and tidy ratios for the same angles.
Sure. Some things work in cycles. Says repeat, tides roll in and out, pendulums swing back and forth. These functions flatten that motion into a line we can measure.
Think about a pendulum swinging back and forth. Suppose to want to put a housing around it. You’re building and old fashioned clock and you don’t want people touching the pendulum. So how much space do you need?
You know that the pendulum hangs down say five inches. So add an inch for safety and call it six inches. But what about left and right?
Well, the pendulum only ever swings up say 30 degrees. So you need the left to right component of thirty degrees, for a five inch pendulum. That’s sine(), the left to right distance spanned by an angle. Multiply that by the five inch pendulum length and you’ve got how wide your case side should be! Add an inch for safety, measure twice, and start cutting!
Any time you see a trig function in an equation, that’s what it is. Someone is at an angle to what is important to you, and you need just the important bit.
Explanations so far are very math-y…
From a mechanical/physical standpoint… (and maybe not strictly accurate but more tangible?)
If you have a vector (speed, force) in some direction, and a baseline reference direction you are interested in (e.g. due east, horizontal)…
The cosine tells you, based on the angle, what portion of your vector is going along your horizontal reference, and the sine is the component orthogonal (vertical in this case)
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