# Double Angle Trigonometric Formulas

DevFeed: [Double Angle Trigonometric Formulas](<https://devfeed.tech/articles/double-angle-trigonometric-formulas-40273.md>)

Original publisher: [Read original article](<https://www.jeremykun.com/2012/05/19/double-angle-trigonometric-formulas/>)

Published: 2012-05-19T23:28:53Z

Content type: tutorial

Language: en

Sources: [Jeremy Kun](<https://devfeed.tech/sources/jeremy-kun.md>)

Topics: [Mathematics](<https://devfeed.tech/topics/mathematics.md>), [Math and Logic](<https://devfeed.tech/topics/math-and-logic.md>)

Tags: [cos](<https://devfeed.tech/tags/cos.md>), [double-angle-identities](<https://devfeed.tech/tags/double-angle-identities.md>), [geometric-transformations](<https://devfeed.tech/tags/geometric-transformations.md>), [linear-algebra](<https://devfeed.tech/tags/linear-algebra.md>), [mathematics](<https://devfeed.tech/tags/mathematics.md>), [matrices](<https://devfeed.tech/tags/matrices.md>), [matrix](<https://devfeed.tech/tags/matrix.md>), [rotation](<https://devfeed.tech/tags/rotation.md>), [trigonometry](<https://devfeed.tech/tags/trigonometry.md>), [vector](<https://devfeed.tech/tags/vector.md>)

## AI overview

This tutorial derives the double-angle sine and cosine identities by representing planar rotations with a 2x2 matrix, squaring that matrix, and comparing it with a rotation by 2θ.

## Source excerpt

Problem: Derive the double angle identities $$\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\\ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$$ Solution: Recall from linear algebra how one rotates a point in the plane. The matrix of rotation (derived by seeing where $ (1,0)$ and $ (0,1)$ go under a rotation by $ \theta$, and writing those coordinates in the columns) is $$A = \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\\ \sin(\theta) & \cos(\theta) \end{pmatrix}$$ Next, note that to rotate a point twice by $ \theta$, we simply multiply the point (as a vector) by $ A$ twice.