The tangent is one of the fundamental trigonometric functions, alongside sine and cosine. It links a triangle’s angles to the ratios of its sides and is indispensable in fields ranging from engineering to physics. In this guide, we’ll walk through the classic right‑triangle definition, illustrate its use with a simple example, and then show how the same value can be derived from other trigonometric functions and calculated using a power‑series expansion.

Step 1: Identify Triangle Components

Label the right triangle so the relationships are clear. Place the right angle at vertex C, making the hypotenuse h opposite this angle. Let the acute angle of interest be θ at vertex A. The side adjacent to θ is labeled b, and the side opposite θ is labeled a. The two legs (a and b) together with the hypotenuse form the complete triangle.

Step 2: Define the Tangent Function

By definition, the tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle:

\[\tan\theta = \frac{a}{b}\]

Step 3: Calculate a Simple Example

Consider an isosceles right triangle, where the legs are equal: a = b. Here, \(\tan\theta = 1\). Since both acute angles are 45°, we confirm that \(\tan45^{\circ}=1\).

Step 4: Derive Tangent from Sine and Cosine

Because \(\sin\theta = \frac{a}{h}\) and \(\cos\theta = \frac{b}{h}\), dividing the two gives: \[\tan\theta = \frac{\sin\theta}{\cos\theta}\]

Step 5: Compute Tangent for Any Angle Using Series Expansion

For higher precision or non‑integer angles, use the Maclaurin series: \[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots\] \[\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots\] Then \[\tan x = \frac{\sin x}{\cos x} = \frac{x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots}{1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots}\]

Truncate the series to the desired accuracy; for most practical purposes, a few terms suffice.