Trigonometry

Some trigonometric formulas.

Basic formulas

\[ \begin{align} \cos(-a)&=\cos a\\ \sin(-a)&=-\sin a\\ \cos(a+b)&= \cos a\cos b-\sin a\sin b \\ \sin(a+b)&=\sin a\cos b+\sin b\cos a\\ \cos^2 a+\sin^2 a&=1\\ \end{align} \]

Tangents and arctangents

\[ \begin{align} \tan(\alpha+\beta)={\tan\alpha+\tan\beta\over1-\tan\alpha\tan\beta}\\ \arctan x+\arctan y = \arctan\left({x+y\over1-xy}\right)\\ \end{align} \]

Cosines of integer multiples as cosines

\[ \begin{align} \cos 2\theta &= \cos^2\theta-\sin^2\theta \\ &=2\cos^2\theta - 1\\ &=1-2\sin^2\theta\\ \cos 3\theta &= 4\cos^3\theta-3\cos\theta \\ \cos 4\theta &= 8\cos^4\theta-8\cos^2\theta+1 \\ \cos 5\theta &= 16\cos^5\theta-20\cos^3\theta+5\cos\theta \\ \end{align} \]

Sines of integer multiples

\[ \begin{align} \sin2\theta&=2\sin\theta\cos\theta\\ \sin3\theta&=-4\sin^3\theta+3\sin\theta \\ \sin4\theta&=-8\sin^3\theta\cos\theta+4\sin\theta\cos\theta \\ \sin5\theta&=16\sin^5\theta-20\sin^3\theta+5\sin\theta\\ \end{align} \] The formula for $\sin5\theta$ is also used in Constructing a pentagon with ruler and compass.

Fractional values

\[ \begin{align} \cos{\theta\over2}&= \sqrt{{1\over2}(1+\cos\theta)} \\ \sin{\theta\over2}&= \sqrt{{1\over2}(1-\cos\theta)} \\ \end{align} \]

Particular values

0 and π

\[ \begin{align} \cos{0}&=1\\ \sin{0}&=0\\ \cos{\pi}&=-1\\ \sin{\pi}&=0\\ \end{align} \]

Halves

\[ \begin{align} \cos{\pi\over 2}&=0\\ \sin{\pi\over 2}&=1\\ \end{align} \]

Thirds

\[ \begin{align} \cos{\pi\over3}&={1\over2} \\ \sin{\pi\over3}&={\sqrt3\over2} \\ \cos{2\pi\over3}&=-{1\over2} \\ \sin{2\pi\over3}&={\sqrt3\over2} \\ \end{align} \]

Quarters

\[ \begin{align} \cos{\pi\over 4}&=\frac1{\sqrt 2}\\ \sin{\pi\over 4}&=\frac1{\sqrt 2}\\ \end{align} \]

Fifths

\[ \begin{align} \cos{\pi\over5} &=\cos{36^\circ} &={1+\sqrt5\over4} &={1 \over 2}\phi \\ \cos{2\pi\over5} &=\cos{72^\circ} &={\sqrt5-1\over4} &={1 \over 2}(\phi - 1) \\ \cos{3\pi\over5} &=\cos{108^\circ} &={1-\sqrt5\over4} &={1 \over 2}(1-\phi) \\ \end{align} \] where $\phi=(1+\sqrt 5)/2\approx1.618033$, the "Golden ratio".

Tenths

\[ \begin{align} \cos{\pi\over10}&=\cos{18^\circ}&=\sqrt{{5+\sqrt5\over8}} &={1\over 2}\sqrt{2+\phi} \\ \cos{3\pi\over10} &=\cos{54^\circ} &=\sqrt{{5\over8}-{\sqrt5\over8}} \\ \sin{\pi\over10} &=\sin{18^\circ} &={\sqrt 5 - 1\over 4} &= {1 \over 2}(\phi - 1)\\ \end{align} \]

Integrals

\[ \int{1\over\cos\theta}d\theta = \ln(\tan\theta+\sec\theta) + C \] \[ \int{\tan\theta}d\theta = \ln(\sec\theta) + C \]

Mnemonics

Quadrants

In the first, second, third, and fourth quadrants, all, sine, tangent, cosine are positive. The mnemonic for this is "All Sadists Teach Chemistry".

Meanings of sine, cosine and tangent

Sine = Opposite over hypotenuse
Cosine = Adjacent over hypotenuse
Tangent = Opposite over adjacent

The mnemonic for this is "SOH CAH TOA" = sock it to yah.