# what does tangent mean?

I need your help, my question is what does tangent value mean and how can we benefit from it ?

I know that $\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}$, but what does that mean?

Sorry I am not good in mathematics so I need your help.

Well, tangent has a few "meanings", of which $\frac{\sin{\theta}}{\cos{\theta}}$ is but one.

Another important one is the fact that tangent describes slope. If I were to construct an angle with the initial ray on the positive X-axis, and the terminal ray in some direction, the "slope" of the terminal ray is the tangent of the angle. You can also see this in the right triangle that is used for basic trig definitions, where tangent is $\frac{\text{opposite}}{\text{adjacent}}$, which is, in a way, the "slope" of the hypotenuse.

Since you label your question with the tag "circle", I'll give you the circle definition of tangent.

Tangent means that the line touches the circle (or other curve) at exactly one point. The wikipedia page for tangent actually has a great image (right side, third image down) showing a tangent as compared to a secant and chord, two other circle terms that are important to know.

http://en.wikipedia.org/wiki/Tangent

Functionally, hexafraction's comment that the tangent describes "slope" is applicable here too. The slope of the tangent line to a circle (or other curve) represents the slope of the curve at the point where the tangent line touches. This is a little abstract, as a point does not have slope, but as a curve doesn't have constant slope, we discuss it's slope at individual points as a measure of how it's moving at that point.

There are things called trig functions where you draw a triangle in the xy plane and take the ratio of certain sides and give these ratios names. If you take an angle in standard position, you can call the side on the x axis (x), the side on the y axis (y) and the hypotenuse (r). Now we can define the trig functions: x/r is called cosine, y/r is called sine and y/x is called tangent. All we are doing is taking certain sides of this triangle and dividing them to get a ratio. So to directly answer your question, tan(x) is (sin(x))/(cos(x)) because ((y/r)/(x/r)) is (y/x) which is tangent.