How do you find the tangent ratio?

Answer 1

In a right angled triangle, the tan of an angle is the ratio of the lengths of the opposite side to the adjacent side.

More generally, on a coordinate plane, draw a circle with its centre at the origin and radius = 1. Draw a line at the required angle from the x-axis, measured in an anticlockwise direction. Suppose this line meets the unit circle at the point whose coordinates are (a,b). Then the tan of the angle is b/a. Make sure that both a and b have their appropriate signs.

An analytical definitions of tan(x), where x is an angle measured in radians does exist but it is rather complicated - and limitations of this browser make it even more difficult to explain. However, using the identity

tan(x) = sin(x) /cos(x)

tan(x) can be derived as the ratio of sine and cosine, where these are defined as the following infinite sums:

sin(x) = x/1! - x3/3! + x5/5! - x7/7! + ...

and

cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...