Identify and label the parts of a right triangle. The right angle will be at vertex C, and the side opposite it will be the hypotenuse h. The angle θ will be at vertex A, and the remaining vertex will be B. The side adjacent to angle θ will be side b and the side opposite angle θ will be side a. The two sides of a triangle that are not the hypotenuse are known as the legs of the triangle.

Define the tangent. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In the case of the triangle in Step 1, tan θ = a/b.

Determine the tangent for a simple right triangle. For example, the legs of an isosceles right triangle are equal, so a/b = tan θ = 1. The angles are also equal so θ = 45 degrees. Therefore, tan 45 degrees = 1.

Derive the tangent from the other trigonometric functions. Since sine θ = a/h and cosine θ = b/h, then sine θ / cosine θ = (a/h) / (b/h) = a/b = tan θ. Therefore, tan θ = sine θ / cosine θ.

Calculate the tangent for any angle and desired accuracy:

sin x = x - x^3/3! + x^5/5! - x^7/7! + ... cosine x = 1 - x^2/2! + x^4/4! - x^6/6! + ... So tan x = (x - x^3/3! + x^5/5! - x^7/7! + ...) / (1 - x^2/2! + x^4/4! - x^6/6! + ...)