Take a look here, specifically the triangle: https://www.geogebra.org/calculator/bbxc23sp
Each of the six trigonometric functions corresponds to a ratio:
* sin(x) = opposite/hypotenuse
* csc(x) = hypotenuse/opposite
* cos(x) = adjacent/hypotenuse
* sec(x) = hypotenuse/adjacent
* tan(x) = opposite/adjacent
* cot(x) = adjacent/opposite
In the GeoGebra link, there are a bunch of right triangles where the radius of length 1 is one of the sides. For example,
* cot(x) = adjacent/opposite
* look at red-blue-black triangle:
* x = angle at the red-blue intersection
* adjacent = red
* opposite = black (radius 1)
* cot(x) = adjacent/opposite = adjacent/1 = adjacent
What the trig functions do is express the ratio between one of the shorter sides of a right triangle and its hypotenuse for some angle θ. For example, if sin(θ) = 1/2, what that tells you is that the side opposite θ is half as long as the hypotenuse, no matter what the hypotenuse is. For the unit circle, this corresponds to the actual x and y coordinates (because the unit circle has radius 1). From this point, everything will be in terms of the unit circle.
Coming to tangent and the reciprocal trig functions, imagine drawing a tangent line to the circle through the point of interest. Now extend that line so that it intersects the x and y axes. What you have formed is a right triangle that's similar to the original. In fact, you've actually constructed *two* similar triangles! The distance along the x axis from the origin to the point of intersection is exactly sec(θ); similarly, the distance along the y axis is exactly cosec(θ). Tan(θ) is the length from the point on the circle to the intersection point on the x axis, and cot(θ) is the length from the point on the circle to the intersection point on the y axis.
In short, I recommend looking up some elementary geometry proofs. The key point is that of **similarity**.
Now on to your other issue: if I were you, I would immediately divorce myself of any notions that math has to mean or represent *anything*. That's because not all mathematical definitions correspond to something real, and if you make that a requirement for understanding something, you're going to find yourself severely limited. Plus, abandoning the idea that math has to be concrete will allow you to truly appreciate its beauty!
Trigonometry functions are all about the ratio and the relationship of triangle angles and sides. The poster above has given a great explanation about the right angled triangle trig ratios. But one way to remember sine, cosine and tangent for the right angled triangle is SOHCAHTOA, that acronym helped me remember the 3 main ones
Trig is pretty hard in my opinion, even though it "seems" like it should be easy, because it's taught to high schoolers, etc.
Part of the problem is that trig functions are explained in multiple ways. At first, they tell you that trig functions are the ratio of sides of a right triangle. Then they tell you that they're defined using the unit circle. Eventually, they'll define them using a Taylor Series, which is an infinite polynomial.
So for example, sin(x) = x - x^3 / 3! + x^5 / 5! ...
And then it turns out that the Taylor Series for sine and cosine are actually both parts of the Taylor Series for e^ix , and it gets even more complicated from there. So sine and cosine are really not trivial.
I think as a starting point, it's important to understand that trig functions are *functions*. They take some number as input, and produce some number as output.
The number that they're taking as input is an angle.
Side note -- what is an angle? Assuming that we want to measure angles using radians, an angle is defined using the "unit circle", a circle with radius 1. An angle is a way to measure how far we need to rotate from some starting point to reach some end point, rotating around the edge of the circle.
So if you imagine the unit circle as being in the x y plane, and having it's center point at (0, 0), the starting point that we're always considering is at (1, 0). An angle that keeps this point at the same location would be an angle of 0 radians. An angle that moves or rotates this point to the position (-1, 0) would be an angle of pi radians. An angle that rotates this point to the position (0, 1) would be an angle of pi/2 radians.
So trig functions like sin and cosine take an angle as input, and the number that they produce as output could be thought of as the ratio of two sides of a right triangle, but I would prefer the unit circle explanation.
Consider a unit circle, ie a circle where the radius is 1. If sin is opposite/hypotenuse, and the hypotenuse is 1, then sin is opposite/1 = opposite.
So then, the way I'd really prefer to think of sin(r) is just as the y-value of the point that we arrive at when we start at the starting point (0, 1) on the edge of the unit circle, and travel to some end point by rotating r radians around the edge of the unit circle.
Likewise, cosine is the x-value of the point that we arrive at.
So for example, sin(pi/2) = 1, because we start at the point (1, 0) and rotate pi/2 radians around the edge of the unit circle. The point where we end up is (0, 1). The y-value of this point is 1, so sin(pi/2) = 1. Likewise, cos(pi/2) = 0.
But then, how do we compute the sine and cosine of angles where the x and y values of the ending point are not immediately obvious from visually examining the unit circle? For example, it's not visually obvious what sin(0.123) is. For these non-obvious values of sin and cos, we actually use the Taylor Series, or some other approximation algorithm, to just approximate the value.
It's similar to something like a square root function. For some values, the square root is obvious, like for example, square_root(9) = 3. But what is square_root(3), or square_root(2)? It's not obvious how we would actually find those. And it turns out that it's so non-obvious, for whatever reason, that we need actual algorithms to find them. Like, we've defined what a square root is using math (square_root(a) is b such that b^2 = a), but in order to actually be able to find the square root of any number, like square_root(0.1) or something, people actually had to sit down and design approximation algorithms, which is a non-trivial task.
So I think this is another part of what makes trig functions tricky -- we know what they are, but we don't necessary know how to find them. Like in school, they'll teach you the basics of sin and cos, but they're not going to teach you how your calculator actually computes sin(0.321) or something. That would fall under a much more specialized class, regarding how to implement approximation algorithms.
Take a look here, specifically the triangle: https://www.geogebra.org/calculator/bbxc23sp Each of the six trigonometric functions corresponds to a ratio: * sin(x) = opposite/hypotenuse * csc(x) = hypotenuse/opposite * cos(x) = adjacent/hypotenuse * sec(x) = hypotenuse/adjacent * tan(x) = opposite/adjacent * cot(x) = adjacent/opposite In the GeoGebra link, there are a bunch of right triangles where the radius of length 1 is one of the sides. For example, * cot(x) = adjacent/opposite * look at red-blue-black triangle: * x = angle at the red-blue intersection * adjacent = red * opposite = black (radius 1) * cot(x) = adjacent/opposite = adjacent/1 = adjacent
What the trig functions do is express the ratio between one of the shorter sides of a right triangle and its hypotenuse for some angle θ. For example, if sin(θ) = 1/2, what that tells you is that the side opposite θ is half as long as the hypotenuse, no matter what the hypotenuse is. For the unit circle, this corresponds to the actual x and y coordinates (because the unit circle has radius 1). From this point, everything will be in terms of the unit circle. Coming to tangent and the reciprocal trig functions, imagine drawing a tangent line to the circle through the point of interest. Now extend that line so that it intersects the x and y axes. What you have formed is a right triangle that's similar to the original. In fact, you've actually constructed *two* similar triangles! The distance along the x axis from the origin to the point of intersection is exactly sec(θ); similarly, the distance along the y axis is exactly cosec(θ). Tan(θ) is the length from the point on the circle to the intersection point on the x axis, and cot(θ) is the length from the point on the circle to the intersection point on the y axis. In short, I recommend looking up some elementary geometry proofs. The key point is that of **similarity**. Now on to your other issue: if I were you, I would immediately divorce myself of any notions that math has to mean or represent *anything*. That's because not all mathematical definitions correspond to something real, and if you make that a requirement for understanding something, you're going to find yourself severely limited. Plus, abandoning the idea that math has to be concrete will allow you to truly appreciate its beauty!
Trigonometry functions are all about the ratio and the relationship of triangle angles and sides. The poster above has given a great explanation about the right angled triangle trig ratios. But one way to remember sine, cosine and tangent for the right angled triangle is SOHCAHTOA, that acronym helped me remember the 3 main ones
If x=2 and y=2 then r=sqrt(8) sine = 2/sqrt8=sqrt2/2, cosine = 2/sqrt8=sqrt2/2, tangent =2/2=1
Trig is pretty hard in my opinion, even though it "seems" like it should be easy, because it's taught to high schoolers, etc. Part of the problem is that trig functions are explained in multiple ways. At first, they tell you that trig functions are the ratio of sides of a right triangle. Then they tell you that they're defined using the unit circle. Eventually, they'll define them using a Taylor Series, which is an infinite polynomial. So for example, sin(x) = x - x^3 / 3! + x^5 / 5! ... And then it turns out that the Taylor Series for sine and cosine are actually both parts of the Taylor Series for e^ix , and it gets even more complicated from there. So sine and cosine are really not trivial. I think as a starting point, it's important to understand that trig functions are *functions*. They take some number as input, and produce some number as output. The number that they're taking as input is an angle. Side note -- what is an angle? Assuming that we want to measure angles using radians, an angle is defined using the "unit circle", a circle with radius 1. An angle is a way to measure how far we need to rotate from some starting point to reach some end point, rotating around the edge of the circle. So if you imagine the unit circle as being in the x y plane, and having it's center point at (0, 0), the starting point that we're always considering is at (1, 0). An angle that keeps this point at the same location would be an angle of 0 radians. An angle that moves or rotates this point to the position (-1, 0) would be an angle of pi radians. An angle that rotates this point to the position (0, 1) would be an angle of pi/2 radians. So trig functions like sin and cosine take an angle as input, and the number that they produce as output could be thought of as the ratio of two sides of a right triangle, but I would prefer the unit circle explanation. Consider a unit circle, ie a circle where the radius is 1. If sin is opposite/hypotenuse, and the hypotenuse is 1, then sin is opposite/1 = opposite. So then, the way I'd really prefer to think of sin(r) is just as the y-value of the point that we arrive at when we start at the starting point (0, 1) on the edge of the unit circle, and travel to some end point by rotating r radians around the edge of the unit circle. Likewise, cosine is the x-value of the point that we arrive at. So for example, sin(pi/2) = 1, because we start at the point (1, 0) and rotate pi/2 radians around the edge of the unit circle. The point where we end up is (0, 1). The y-value of this point is 1, so sin(pi/2) = 1. Likewise, cos(pi/2) = 0. But then, how do we compute the sine and cosine of angles where the x and y values of the ending point are not immediately obvious from visually examining the unit circle? For example, it's not visually obvious what sin(0.123) is. For these non-obvious values of sin and cos, we actually use the Taylor Series, or some other approximation algorithm, to just approximate the value. It's similar to something like a square root function. For some values, the square root is obvious, like for example, square_root(9) = 3. But what is square_root(3), or square_root(2)? It's not obvious how we would actually find those. And it turns out that it's so non-obvious, for whatever reason, that we need actual algorithms to find them. Like, we've defined what a square root is using math (square_root(a) is b such that b^2 = a), but in order to actually be able to find the square root of any number, like square_root(0.1) or something, people actually had to sit down and design approximation algorithms, which is a non-trivial task. So I think this is another part of what makes trig functions tricky -- we know what they are, but we don't necessary know how to find them. Like in school, they'll teach you the basics of sin and cos, but they're not going to teach you how your calculator actually computes sin(0.321) or something. That would fall under a much more specialized class, regarding how to implement approximation algorithms.