Pythagorean Identity in Trigonometry

What is the Pythagorean Identity?

The Pythagorean Identity is a fundamental trigonometric identity that relates the squares of sine and cosine functions to the constant 1. It is derived from the Pythagorean theorem and forms the basis for many other trigonometric identities.

The Three Forms of Pythagorean Identity

There are three main forms of the Pythagorean Identity:

1. \(\sin^2 \theta + \cos^2 \theta = 1\)
2. \(1 + \tan^2 \theta = \sec^2 \theta\)
3. \(1 + \cot^2 \theta = \csc^2 \theta\)

Where:

• \(\theta\) is any angle
• \(\sin\), \(\cos\), \(\tan\), \(\cot\), \(\sec\), and \(\csc\) are trigonometric functions

Derivation and Explanation

The first form of the Pythagorean Identity (\(\sin^2 \theta + \cos^2 \theta = 1\)) can be derived directly from the Pythagorean theorem applied to a right triangle inscribed in a unit circle:

Consider a right triangle with hypotenuse 1 (radius of the unit circle):

• The opposite side length is \(\sin \theta\)
• The adjacent side length is \(\cos \theta\)

Applying the Pythagorean theorem:

\[ \begin{align*} (\text{opposite})^2 + (\text{adjacent})^2 &= (\text{hypotenuse})^2 \\ (\sin \theta)^2 + (\cos \theta)^2 &= 1^2 \\ \sin^2 \theta + \cos^2 \theta &= 1 \end{align*} \]

Example Calculation

Let's verify the Pythagorean Identity for \(\theta = 30°\):

\[ \begin{align*} \sin^2 30° + \cos^2 30° &= (\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2 \\[10pt] &= \frac{1}{4} + \frac{3}{4} \\[10pt] &= 1 \end{align*} \]

Visual Representation

This diagram illustrates a 30-60-90 triangle inscribed in the unit circle, visually representing the Pythagorean Identity for \(\theta = 30°\).