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Trigonometric Functions

The Trigonometric Functions are functions of angles

• they relate angles of a triangle to the length of its sides

• $\sin \theta$ = length of edge, opposite to $\theta$
• $\cos \theta$ = length of edge, adjacent to $\theta$
• when the hypotenuse is 1, by Pythagoras Theorem, we get $\cos^2 \alpha + \sin^2 \alpha = 1$

Unit circle:

Other trigonometric functions are derives from sine and cosine:

• tangent: $\tan \alpha = \cfrac{\sin \alpha}{\cos \alpha}$
• cotangent: $\cot \alpha = \cfrac{\cos \alpha}{\sin \alpha}$
• secant: $\sec \alpha = \cfrac{1}{\cos \alpha}$
• co-secant: $\csc \alpha = \cfrac{1}{\sin \alpha}$

Inverse Trigonometric functions:

• arcsin: $\arcsin \alpha = \sin^{-1} \alpha$
• arccos: $\arccos \alpha = \cos^{-1} \alpha$
• both have restricted domain $[-1, 1]$ because $\cos$ and $\sin$ can output only $[-1, 1]$

(Source: [1])

Taylor Expansion

• $\cos x = \sum\limits_{k=0}^\infty (-1)^k \cfrac{x^{2k}}{(2k)!}$
• $\sin x = \sum\limits_{k=0}^\infty (-1)^k \cfrac{x^{2k + 1}}{(2k + 1)!}$
• relates Exponential Function and Trigonometric Functions
• $e^{ix} = \cos x + i \sin x$ where $i = \sqrt {-1}$

With Taylor Expansions, we can pretend that $\sin x$, $\cos x$ and Exponential are long Polynomial Functions

Trigonometric Identities

• $\alpha$ radians = $180\, \alpha\, /\, \pi$ degrees
• $\beta$ degrees = $\pi\, \beta\, /\, 180$ radians

Fundamental

• $\sin (-\alpha) = -\sin \alpha$
• $\cos (-\alpha) = \cos \alpha$
• $\sin \left(\pi/2 - \alpha\right) = \cos \alpha$
• $\cos \left(\cfrac{\pi}{2} - \alpha \right) = \sin \alpha$
• $\sin^2 \alpha + \cos^2 \alpha = 1$
• $\tan \alpha = \cfrac{\sin \alpha}{\cos \alpha}$
• $\cot \alpha = \cfrac{\cos \alpha}{\sin \alpha} = \cfrac{1}{\tan \alpha}$

Double angles

• $\sin 2\alpha = 2\, \sin \alpha \, \cos \alpha$
• $\cos 2\alpha = 2\, \cos^2 \alpha - 1 = 1 - 2\, \sin^2 \alpha = \cos^2 \alpha - sin^2 \alpha$
• $\tan 2\alpha = \cfrac{2\, \tan \alpha}{1 - \tan^2 \alpha}$

Sums

• $\sin \alpha + \sin \beta = 2\, \sin\cfrac{\alpha +\beta}{2}\, \cos\cfrac{\alpha - \beta}{2}$
• $\sin \alpha - \sin \beta = 2\, \cos\cfrac{\alpha +\beta}{2}\, \sin\cfrac{\alpha - \beta}{2}$
• $\cos \alpha + \cos \beta = 2\, \cos\cfrac{\alpha +\beta}{2}\, \cos\cfrac{\alpha - \beta}{2}$
• $\cos \alpha - \cos \beta = 2\, \sin\cfrac{\alpha +\beta}{2}\, \sin\cfrac{\beta - \alpha}{2}$

Products

• $2\, \sin \alpha\, \sin \beta = \cos (\alpha - \beta) - \cos (\alpha + \beta)$
• $2\, \cos \alpha\, \cos \beta = \cos (\alpha - \beta) + \cos (\alpha + \beta)$
• $2\, \sin \alpha\, \cos \beta = \sin (\alpha + \beta) + \sin (\alpha - \beta)$

Sums Inside Trigs

• $\sin (\alpha \pm \beta) = \sin \alpha\, \cos \beta \pm \sin \beta\, \cos \alpha$
• $\cos (\alpha \pm \beta) = \cos \alpha\, \cos \beta \mp \sin \alpha\, \sin \beta$
• $\tan (\alpha \pm \beta) = \cfrac{\tan \alpha \pm \tan\beta}{1 \mp \tan \alpha\, \tan \beta}$

Secs and Cosecs

• $\sec \alpha = \cfrac{1}{\cos \alpha}$
• $\csc \alpha = \cfrac{1}{\sin \alpha}$
• $1 + \tan^2 \alpha = \sec^2 \alpha$
• $\tan \left(\cfrac{\pi}{2} - \alpha \right) = \cot \alpha$

Inverse Trigs

• $\sin (\arcsin \alpha) = \alpha$
• $\cos (\arccos \alpha) = \alpha$
• $\tan (\arctan \alpha) = \alpha$
• $\arcsin (\sin \alpha) = \alpha$
• $\arccos (\cos \alpha) = \alpha$
• $\arctan (\tan \alpha) = \alpha$

Tangents

• $\tan^2 \alpha = \cfrac{1 - \cos \alpha}{1 + \cos \alpha}$
• $\tan \cfrac{2}{\alpha} = \cfrac{2\, \tan \alpha}{1 - \tan^2 \alpha}$
• $\tan \cfrac{\alpha}{2} = \cfrac{\sin \alpha}{1 + \cos \alpha} = \cfrac{1 - \cos \alpha}{\sin \alpha}$
• $\tan \cfrac{\alpha + \beta}{2} = \cfrac{\sin \alpha + \sin \beta}{\cos \alpha + \cos \beta}$
• $\tan \left(\cfrac{\alpha}{2} + \cfrac{\pi}{4}\right) = \tan \alpha + \sec \alpha = \cfrac{1 + \sin \alpha}{\cos \alpha}$
• $\tan 3\alpha = \cfrac{3\, \tan \alpha - \tan^3 \alpha}{1 - 3\, \tan^3 \alpha}$

Derivatives and Integrals

Derivatives

• $\sin' x = \cos x$
• $\cos' x = -\sin x$
• $\tan' x = \sec^2 x$
• $\big(\sin^2 x \big)' = 2\, \sin x\, \cos x = \sin 2x$
• $\big(\cos^2 x \big)' = -2\, \cos x\, \sin x = -\sin 2x$
• $\big(\tan^2 x \big)' = 2\, \tan x + 2\, \tan^3 x$
• $\csc' x = -\cot x\, \csc x$
• $\sec' x = \tan x\, \sec x$
• $\cot' x = -\csc^2 x$

Integrals

Basic Integrals:

• $\int \sin x\, dx = -\cos x + C$
• $\int \cos x\, dx = \sin x + C$
• $\int \tan x\, dx = -\ln | \cos x | + C$
• $\int \sec x\, dx = \ln | \tan x + \sec x | + C$
• $\int \csc x\, dx = -\ln | \cot x + \csc x | + C$
• $\int \cot x\, dx = \ln | \sin x | + C$
• $\int \sin^2 x\, dx = \cfrac{x}{2} - \cfrac{\sin 2x}{4} + C$
• $\int \cos^2 x\, dx = \cfrac{x}{2} + \cfrac{\sin 2x}{4} + C$
• $\int \sin x\, \cos x\, dx = -\cfrac{1}{2}\, \cos^2 x + C$

Harder Integrals:

• $\int x\, \sin x\, dx = \sin x - x\, \cos x + C$
• $\int x\, \cos x\, dx = \cos x + x\, \sin x + C$
• $\int \cfrac{1}{1 + \sin x}\, dx = \tan \left( \cfrac{x}{2} - \cfrac{\pi}{4} \right) + C$
• $\int \cfrac{1}{1 + \cos x} dx = \tan \left( \cfrac{x}{2} \right) + C$