Trigonometry’s summary of the properties and applications: remarkable transformations, equations, formulas, functions, remarkable values
Trigonometry is a branch of mathematics that studies the relationship between angles and the lengths of the sides of a triangle. It uses mathematical functions such as sine, cosine and tangent to describe these relationships. Trigonometry is used in many fields such as geometry, astronomy, physics, mechanics, and engineering.
Definition field
\(cos(x)\) and \(arccos(x)\) set to \([-1;1]\)
\(sin(x)\) and \(arcsin(x)\) set to \([-1;1]\)
\(arctan(x)\) set to \(\mathbb{R} \rightarrow\) image set to \(]-\frac{\pi}{2};\frac{\pi}{2}[\)
Fundamental relationships
$$tan(x)=\frac{sin(x)}{cos(x)}$$ $$sin^2(x) + cos^2(x) = 1 $$ $$sin^2(x) = \frac{tan^2(x)}{1+tan^2(x)}$$ $$cos^2(x) = \frac{1}{1+tan^2(x)}$$
Remarkable transformations
| $$sin(2\pi+x) = sin(x)$$ | $$cos(2\pi+x) = cos(x)$$ | $$tan(2\pi+x) = tan(x)$$ |
| $$sin(-x) = -sin(x)$$ | $$cos(-x) = cos(x)$$ | $$tan(-x) = -tan(x)$$ |
| $$sin(\pi-x) = sin(x)$$ | $$cos(\pi-x) = -cos(x)$$ | $$tan(\pi-x) = -tan(x)$$ |
| $$sin(\pi+x) = -sin(x)$$ | $$cos(\pi+x) = -cos(x)$$ | $$tan(\pi+x) = tan(x)$$ |
| $$sin(\frac{\pi}{2}-x) = cos(x)$$ | $$cos(\frac{\pi}{2}-x) = sin(x)$$ | $$tan(\frac{\pi}{2}-x) = \frac{1}{tan(x)}$$ |
| $$sin(\frac{\pi}{2}+x) = cos(x)$$ | $$cos(\frac{\pi}{2}+x) = sin(x)$$ | $$tan(\frac{\pi}{2}+x) = \frac{-1}{tan(x)}$$ |
| $$sin(\frac{3\pi}{2}-x) = -cos(x)$$ | $$cos(\frac{3\pi}{2}-x) = -sin(x)$$ | $$tan(\frac{3\pi}{2}-x) = \frac{1}{tan(x)}$$ |
| $$sin(\frac{3\pi}{2}+x) = -cos(x)$$ | $$cos(\frac{3\pi}{2}+x) = sin(x))$$ | $$tan(\frac{3\pi}{2}+x) = \frac{-1}{tan(x)}$$ |
| $$cos(x-\frac{\pi}{2}) = sin(x))$$ |
Trigonometric equations
\(k \in \mathbb{Z}\)
If \(sin(a)=sin(b)\), then \(a=b+2k\pi\) or \(a=\pi-b+2k\pi\)
If \(cos(a)=cos(b)\), then \(a=b+2k\pi\) or \(a=-b+2k\pi\)
If \(tan(a)=tan(b)\), then \(a=b+2k\pi\)
Addition formulas
| $$sin(a+b) = sin(a).cos(b) + sin(b).cos(a)$$ | $$cos(a+b) = cos(a).cos(b) – sin(a).sin(b)$$ |
| $$sin(a-b) = sin(a).cos(b) – sin(b).cos(a)$$ | $$cos(a-b) = cos(a).cos(b) + sin(a).sin(b)$$ |
| $$tan(a+b) = \frac{tan(a)+tan(b)}{1-tan(a).tan(b)}$$ | $$tan(a-b) = \frac{tan(a)-tan(b)}{1-tan(a).tan(b)}$$ |
| $$sin(p)+sin(q) = 2.sin(\frac{p+q}{2}).cos(\frac{p-q}{2})$$ | $$sin(p)-sin(q) = 2.sin(\frac{p-q}{2}).cos(\frac{p+q}{2})$$ |
| $$cos(p)+cos(q) = 2.cos(\frac{p+q}{2}).cos(\frac{p-q}{2})$$ | $$cos(p)-cos(q) = -2.sin(\frac{p+q}{2}).sin(\frac{p-q}{2})$$ |
| $$tan(p)+tan(q) = \frac{sin(p+q)}{cos(p).cos(q)}$$ | $$tan(p)-tan(q) = \frac{sin(p-q)}{cos(p).cos(q)}$$ |
| $$sin(a).sin(b)=\frac{1}{2}*(cos(a-b)-cos(a+b))$$ | $$cos(a).cos(b)=\frac{1}{2}*(cos(a+b)-cos(a-b))$$ |
| $$sin(a).cos(b)=\frac{1}{2}*(sin(a+b)-sin(a+b))$$ |
Duplicate formulas
$$sin(2a)=2 sin(a).cos(a)=\frac{tan(a)}{1+tan^2(a)}$$ $$sin^2(x)+cos^2(x)=1$$ $$cos(2a) = cos^2(a)-sin^2(a) = 2cos^2(a)-1 = 1-2sin^2(a)$$ $$tan(2a) = \frac{2tan(a)}{1-tan^2(a)}$$ $$sin^2(a) = \frac{1-cos(2a)}{2}$$ $$cos^2(a) = \frac{1+cos(2a)}{2}$$ $$tan^2(a) = \frac{1-cos(2a)}{1+cos(2a)}$$ $$tan(a)= \frac{sin(2a)}{1+cos(2a)}=\frac{1-cos(2a)}{sin(2a)}$$ By setting \(t=tan (\frac{a}{2})\): $$sin(a)=\frac{2t}{1+t^2}$$ $$cos(a)=\frac{1-t^2}{1+t^2}$$ $$tan(a)=\frac{2t}{1-t^2}$$
De Moivre’s formula
$$(cos(a)+i.sin(a))^n=cos(n.a)+i.sin(n.a)$$
Euler formula
$$cos(\theta)=\frac{1}{2}(e^{i.\theta}+e^{-i.\theta})$$ $$sin(\theta)=\frac{1}{2i}(e^{i.\theta}-e^{-i.\theta})$$ $$e^i\theta=cos(\theta)+i.sin(\theta)$$
Remarkable point values
| $$0$$ | $$\frac{\pi}{6}$$ | $$\frac{\pi}{4}$$ | $$\frac{\pi}{3}$$ | $$\frac{\pi}{2}$$ | |
| $$sin(x)$$ | $$0$$ | $$\frac{1}{2}$$ | $$\frac{\sqrt{2}}{2}$$ | $$\frac{\sqrt{3}}{2}$$ | $$1$$ |
| $$cos(x)$$ | $$1$$ | $$\frac{\sqrt{3}}{2}$$ | $$\frac{\sqrt{2}}{2}$$ | $$\frac{1}{2}$$ | $$0$$ |
| $$tan(x)$$ | $$0$$ | $$\frac{\sqrt{3}}{3}$$ | $$1$$ | $$\sqrt{3}$$ | $$\nexists$$ |
| $$cotan(x)$$ | $$\nexists$$ | $$\sqrt{3}$$ | $$1$$ | $$\frac{\sqrt{3}}{3}$$ | $$0$$ |
Trigonometric functions
Direct circulars: \(sin(x)\), \(cos(x)\), \(tan(x)\), \(cotan(x)\), \(sec(x)\), \(cosec(x)\)
$$cotan(x) = \frac{1}{tan(x)}$$
$$cotanh(x) = \frac{1}{tanh(x)}$$
$$sec(x) = \frac{1}{cos(x)}$$
$$sec(x) = \frac{1}{cosh(x)}$$
$$cosec(x) = \frac{1}{sin(x)}$$
$$cotech(x) = \frac{1}{sinh(x)}$$
Reciprocal circulars: \(arcsin(x)\), \(arccos(x)\), \(arctan(x)\), \(arccotan(x)\), \(arcsec(x)\), \(arccosec(x)\)
Direct hyperbolics: \(sinh(x)\), \(cosh(x)\), \(tanh(x)\), \(cotanh(x)\), \(sech(x)\), \(cosech(x)\)
Reciprocal hyperbolics: \(argsinh(x)\), \(argcosh(x)\), \(argtanh(x)\), \(argcotanh(x)\), \(argsech(x)\), \(argcosech(x)\)