Sine is an odd function
and cosine an even function.
The cosine of an angle is equal to the sine of its co mplement.
\(\sin(\theta) = \cos\bigl(90°\!-\theta\bigr)\)
\(\cos(\theta) = \sin\bigl(90°\!-\theta\bigr)\)
\(\cos(-\theta) = \cos(\theta)\)
\(\sin(-\theta) = -\sin(\theta)\)
The Pythagorean identities
follow from sine and cosine being
the lengths legs of a right triangle in the unit circle.
\(\sin^2(\theta) + \cos^2(\theta) = 1\)
\(\tan^2(\theta) + 1 = \sec^2(\theta)\)
\(1 + \cot^2(\theta) = \csc^2(\theta)\)
The sum-of-angles formulas can be derived geometrically,
and can be used as the basis for all the other following formulas.
\(\displaystyle \cos(\theta+\varphi) = \cos(\theta)\cos(\varphi) - \sin(\theta)\sin(\varphi)\)
\(\displaystyle \sin(\theta+\varphi) = \sin(\theta)\cos(\varphi) + \cos(\theta)\sin(\varphi)\)
\(\displaystyle \tan(\theta+\varphi) = \frac{\tan(\theta)+\tan(\varphi)}{1 - \tan(\theta)\tan(\varphi)}\)
The double-angle formulas and half-angle formulas
follow from the sum-of-angles formulas and the first Pythagorean identity.
\(\phantom{\sqrt{\biggl({\frac{a}{b}\biggr)}}} \displaystyle \sin(2\theta) = 2\cos(\theta)\sin(\theta) \)
\(\displaystyle \cos(2\theta) = \cos^2(\theta)-\sin^2(\theta) \phantom{\sqrt{\biggl({\frac{a}{b}\biggr)}}} \)
\(\displaystyle \sin\biggl(\frac{\theta}{2}\biggr) = \sqrt{\frac{1-\cos(\theta)}{2}}\)
\(\displaystyle \cos\biggl(\frac{\theta}{2}\biggr) = \sqrt{\frac{1+\cos(\theta)}{2}}\)
Extracurricular bonus fact: the trigonometric functions can be defined as complex functions
in terms of the exponential constant \({\mathrm{e} \approx 2.71828182845904}\) and the imaginary unit \(\mathfrak{i} = \sqrt{-1}\).
This is the algebraic basis of the idea that “complex geometry is about rotations.”
\(\displaystyle \mathrm{e}^{\mathfrak{i}\theta} = \cos(\theta) + \mathfrak{i}\sin(\theta) \)
\(\displaystyle \cos(\theta) = \frac{\mathrm{e}^{\mathfrak{i}\theta} + \mathrm{e}^{-\mathfrak{i}\theta}}{2} \)
\(\displaystyle \sin(\theta) = \frac{\mathrm{e}^{\mathfrak{i}\theta} - \mathrm{e}^{-\mathfrak{i}\theta}}{2\mathfrak{i}} \)