Question about tan, cos, and sin

[Spiralshell]

So, I am trying to understand what tangent, cosine, and sine are in practical terms e.g. if something is tangent with something else we know it is touching that something. That seems like a practical definition of tangent to me.

So, then what is a sine? what is cosine?

I was thinking like this: if sine is y on the cartesian plane then it is something raised or lowered and if cosine is x then it is something being moved left or right. Then if tangent is sine/cosine it must be...raised and lowered at the same time? if it is being raised and lowered on all sides, it is touching all sides..?

Also, does anyone know if sine, cosine, and tangent were created to do or solve something specific or was it just discovered? If I understand the first part of my question (what sine, cosine, and tangent all mean without a mathematical explanation) then this part will be answered.

I am sorry if this sounds really confusing,

 

[Number Nine]

Er...not quite. We do say that a line is "tangent" to something, but this is not the same as the tangent function (though it is related in a specific sense)). All of the trigonometric function are defined very nicely in terms of right triangles inside the unit circle...

http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg

 

[hondaman520]

yeah I think you may be misunderstanding the context of tangent. Often in calculus a tangent line is used to represent instantaneous rate of change at a given point.

The trig functions sine cosine and tangent are different, they refer to the relations of a right triangle on the coordinate plane, and can also be represented on a graph as waves, (often used in quantom physics).

For me, the best way to visualize these functions is this for example:

for sine of something (x) = 3. or in math language: sinx=3

you can picture this function as x is the measure (in degrees or radians) of the angle on the coordinate plane, starting from 0, 360 degrees.

3, represents the RATIO of the side opposite of the angle(X) over the hypotenuse.

regardless of what angle x is, you have this information: y=3, and r(radius/hypotenuse) = 1. In fact, you can find x by punching arcsin(3) in a calculator, to find angle measure x. These bits of information are inverse from each other.

These trig functions are simply ratios of two sides of a right triangle.

Cosine is the ratio o the adjacent side over the hypotenuse.. and so on...

These functions are often used to find more information about side length of a right triangle. You can visualize inverse trigg functions as just the opposite, these will give you information about angle measure of that triangle. This information is really abstract which makes it boring and hard to visualize. but ultimately, sin, cos, or tan of something will give you the ratio of two sides of a right triangle. (That doesn't necessarily mean a trig function will give you the exact values of the lengths to SCALE size, in real life. But relative to other sides of that triangle, it gives you the exact or extremely closely approximated (via calculator) values.