Can someone tell my what cos, sin and tan actually measure?
In: Mathematics
They measure the ratio of the sides of a right triangle as a function of another angle in the triangle.
Sin and cos are related, as the two non-right-angle angles in a triangle always add to 90. The sin of one angle is the cos of the other angle. The tan is the ratio of the two sides that are not the hypotenuse, and the tan from one angle is 1/tan from the other angle.
Sine and cosine measure the ratio of one side of a right triangle to it’s hypotenuse. Tangent is Sin/Cos. For example Sin(45)=1/sqrt(2) means that the length of the side opposite the 45degree angle divided by the length of the hypotenuse will be 1/sqrt(2). This means that if you know one of the non-90 degree angles of the right triangle and one length or two lengths, you can find the lengths of all three sides and all three angles.
It’s a way of figuring out geometry when you arent given all the facts to start with. It’s important in math, engineering, calculus, physics, construction, etc.
Have a length but need an angle measurement? Trig.
Have an angle but don’t know the lengths of all the connected pieces? Trig.
In geometry these functions, tell you various relations between the length of the side of a triangle with the angles inside them. Basically you can calculate the angles of a triangle, when given the side lengths of a triangle and vice versa.
They are also useful in many other parts of mathematics and physics, which do not directly are related to triangles. They have some connections to circles, you can describe things like water waves or pendulum movements with them and many other things.
On a unit circle (circle with radius 1 and the center at the origin) draw a line from the origin with your desired angle (wrt to the x-axis) as slope. cos is the x component of the point where the line crosses the circle. sin is the y component of the point where the line crosses the circle. tan is the y component of the line when the x component is 1 or -1. the vertical line at 1 or -1 is the tangent of the unit circle with respect to the x-axis.
EDIT: here is a [picture](https://i.imgur.com/U1NTgzA.png). note, tan in the picture uses a slightly different (but equivalent) definition. also [cool video](https://youtu.be/h9CRR_07eAI) from below
Ratios. In a right triangle, sine is the length of the opposite side divided by the length of the hypotenuse, cosine is the length of the adjacent side divided by the length of the hypotenuse, and tangent is the length of the opposite side divided by the adjacent side.
That seems like it would be kind of pointless to do, but if you draw a circle, and use the hypotenuse as a radius, sine and cosine essentially translate polar coordinates (angle and distance) into Cartesian coordinates (x, y).
Take a triangle. If you reduce it and enlarge it, the overall shape doesn’t change, right? Even as one side shrinks or grows, the other sides also shrink and grow to match.
The ratios between each of the three sides is constant and depends only on the angles between the sides. Cos/sin/tan are basically those ratios for a given angle. (Technically, they only work on triangles with a right angle, but you can make any triangle out of two of thoose, so it all works out in practice)
They are essentially a table of ratios for right triangles.
Referencing a corner/angle,
Sin is the ratio between the opposite side and the hypotenuse.
Cos is the ratio between the adjacent side and the hypotenuse.
Tan is the ratio between the opposite side and the adjacent side.
Your calculator just has a huge table of these ratios. Sin(30) has a specific number, because the opposite side divided by the hypotenuse will always be the same no matter how large or small that right triangle with a 30 degree corner is.
I think the unit circle helps to understand a lot of the ideas behind those, and why they come up so often. [https://en.wikipedia.org/wiki/Unit_circle](https://en.wikipedia.org/wiki/Unit_circle)
Everyone is starting explaining stuff with triangles. To me, it’s a lot more obvious looking at this circle. it makes most of the formulas and theorem around them get pretty obvious, instead of something arbitrary you have to learn from memory.
You draw a circle with radius 1. For any given point on that circle, you look at the angle it makes with the horizontal. And then, cosines and sines of that angle are the coordinates of that point on the circle. They are just telling you where you’ll cross the circle if you draw a line starting from the center, and going at that angle.
Granted, Tan is a bit more tricky.
Example of stuff that gets obvious:
_ Sin and cos are always between -1 and 1. Well, the circle just has radius 1, so clearly no point goes further than that.
_ For any x, cos²(x) + sin²(x) = 1. Looks like a weird formula ? That’s just looking at the radius of the circle, putting the coordinates in the formula for distance between the point on the circle and the center. We know that distance is 1, that was the definition.
_ cos is the ratio of adjacent side of a triangle to hypotenuse. You can just draw that triangle in the circle. cosines is the length of that side. Then the division by hypotenuse is fixing the scale, since in our unit circle, hypotenuse is 1. Same for sin.
imagine a circle of radius 1. Now, make an angled line going from the center to the edge of that circle. The sin of the angle of that line is how high up the line goes, and cos is how far left/right the line goes. tangent is the slope of that line.
Latest Answers