Basic Trigonometry

Capturing some basic trigonmetry notes to put them in one place....

Trigonometric functions

Basics

Consider a right triangle and its hypotenuse c, base b, and altitude a. The greek letter theta, $\theta$ is defined to be the angle opposite the altitude, and the greek letter phi, $\varphi$, is defined to be the angle opposite the base.

$\sin \theta =\frac{a}{c}$

$\cos \theta =\frac{b}{c}$

$\tan \theta =\frac{a}{b}$

Re-expressing Pythagoras

${\cos }^2\theta +{\sin }^2\theta =1$

Combining angles

If $\theta$ is the sum or difference of two other angles $\alpha ,\beta$ :

$sin(\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

$sin(\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta$

$cos(\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

$cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta$

Differentiation of trigonometric functions

$\frac{d\sin x}{dx}={\sin }^{\prime }x=\cos x$

$\frac{d\cos x}{dx}={\cos }^{\prime }x=-\sin x$

See also

• Hrabovsky, George; Susskind, Leonard. Classical Mechanics: The Theoretical Minimum (Theoretical Minimum 1) (p. 19). Penguin Books Ltd. Kindle Edition.
• Sin, Cos and Tan of Sum and Difference of Two Angles
• Differentiation of trigonometric functions