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Unit circle

The unit circle is a fundamental tool in mathematics. It is defined as a circle with its center at the origin of the Cartesian plane $\left(0,0\right)$ and a radius equal to $1$.

Properties

The general equation of the unit circle is:

$$x^{2} + y^{2} = 1$$

This means any point $\left(x,y\right)$ on the circle satisfies this relationship.

It intersects the $x-$ and $y-$axes at the points $(1,0)$ $(-1,0)$, $(0,1)$, and $(0,-1)$.

Points on the unit circle are associated with angles measured in radians or degrees. For example:

$0\ ^{\circ} \left(0\ \mathrm{radians}\right): (1,0)$
$90\ ^{\circ} \left(\frac{\pi}{2}\right): (0,1)$
$180\ ^{\circ} \left(\pi\right): (-1,0)$
$270\ ^{\circ} \left(\frac{3\pi}{2}\right): (0,-1)$

Instructions:

Move the slider for $\alpha$ to modify the angle, you will see how the trigonometric functions results change. Below are the formulas for the trigonometric functions.

Definitions

Trigonometric functions are defined using the unit circle or right triangles:

Given an angle $\theta$ in the unit circle, with a corresponding point $(x,y)=(\cos\theta,\sin\theta)$:

Sine $(\sin)$: The $y$-coordinate of the point.

$$\sin\theta = y$$

Cosine $(\cos)$: The $x$-coordinate of the point.

$$\cos\theta = x$$

Tangent $(\tan)$: The ratio of sine to cosine (if $\cos\theta \neq 0$):

$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

Given an acute angle $\theta$ in a right triangle:

Sine: Ratio of the opposite side to the hypotenuse.

$$\sin\theta = \frac{\mathrm{hypotenuse}}{\mathrm{opposite\ cathetus}}$$

Cosine: Ratio of the adjacent side to the hypotenuse.

$$\cos\theta = \frac{\mathrm{hypotenuse}}{\mathrm{adjacent\ cathetus}}$$

Tangent: Ratio of the opposite side to the adjacent side.

$$\tan\theta = \frac{\mathrm{adjacent\ cathetus}}{\mathrm{opposite\ cathetus}}$$

Formulae

$\sin \left( \theta \right) = \frac{\mathrm{opposite\ cathetus}}{\mathrm{hypotenuse}}$
$\cos \left( \theta \right) = \frac{\mathrm{adjacent\ cathetus}}{\mathrm{hypotenuse}}$
$\tan \left( \theta \right) = \frac{\mathrm{opposite\ cathetus}}{\mathrm{adjacent}}$
$\csc \left( \theta \right) = \frac{\mathrm{hypotenuse}}{\mathrm{opposite\ cathetus}}$
$\sec \left( \theta \right) = \frac{\mathrm{hypotenuse}}{\mathrm{adjacent\ cathetus}}$
$\ctg \left( \theta \right) = \frac{\mathrm{adjacent\ cathetus}}{\mathrm{opposite}}$

Trigonometric table

A trigonometric table is a mathematical tool that lists the values of trigonometric functions $(\sin, \cos, \tan, \csc, \sec, \cot)$ for various common angles.

Key Features

Include angles measured in degrees $(0^{\circ},30^{\circ},45^{\circ},60^{\circ},90^{\circ})$ or radians $(0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2})$.
Each cell in the table provides the corresponding value of a trigonometric function for a specific angle. For example:

$\sin 30^{\circ} = \frac{\pi}{6} = 0.5$
$\cos 45^{\circ} = \frac{\pi}{4} = \frac{\sqrt{2}}{2} \approx 0.7071$
$\tan 60^{\circ} = \frac{\pi}{3} = \sqrt{3} \approx 1.732$

$ \theta \left(\ ^{\circ}\right)$	$ \theta (\mathrm{rad})$	$\sin\theta$	$\cos\theta$	$\tan\theta$	$\csc\theta$	$\sec\theta$	$\cot\theta$
$0\ ^{\circ}$	$0$	$0$	$1$	$0$	$\infty$	$1$	$\infty$
$30\ ^{\circ}$	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$2$	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
$45\ ^{\circ}$	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
$60\ ^{\circ}$	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	$2$	$\frac{\sqrt{3}}{3}$
$90\ ^{\circ}$	$\frac{\pi}{2}$	$1$	$0$	$\infty$	$1$	$\infty$	$0$
$120\ ^{\circ}$	$\frac{2 \pi}{3}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	$-2$	$-\frac{\sqrt{3}}{3}$
$135\ ^{\circ}$	$\frac{3 \pi}{4}$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$	$\sqrt{2}$	$-\sqrt{2}$	$-1$
$150\ ^{\circ}$	$\frac{5 \pi}{6}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$	$2$	$-\frac{2\sqrt{3}}{3}$	$-\sqrt{3}$
$180\ ^{\circ}$	$\pi$	$0$	$-1$	$0$	$\infty$	$-1$	$\infty$

\( \theta \left(\ ^{\circ}\right)\)	\( \theta (\mathrm{rad})\)	\(\sin\theta\)	\(\cos\theta\)	\(\tan\theta\)	\(\csc\theta\)	\(\sec\theta\)	\(\cot\theta\)
\(0\ ^{\circ}\)	\(0\)	\(0\)	\(1\)	\(0\)	\(\infty\)	\(1\)	\(\infty\)
\(30\ ^{\circ}\)	\(\frac{\pi}{6}\)	\(\frac{1}{2}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{\sqrt{3}}{3}\)	\(2\)	\(\frac{2\sqrt{3}}{3}\)	\(\sqrt{3}\)
\(45\ ^{\circ}\)	\(\frac{\pi}{4}\)	\(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{2}}{2}\)	\(1\)	\(\sqrt{2}\)	\(\sqrt{2}\)	\(1\)
\(60\ ^{\circ}\)	\(\frac{\pi}{3}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{2}\)	\(\sqrt{3}\)	\(\frac{2\sqrt{3}}{3}\)	\(2\)	\(\frac{\sqrt{3}}{3}\)
\(90\ ^{\circ}\)	\(\frac{\pi}{2}\)	\(1\)	\(0\)	\(\infty\)	\(1\)	\(\infty\)	\(0\)
\(120\ ^{\circ}\)	\(\frac{2 \pi}{3}\)	\(\frac{\sqrt{3}}{2}\)	\(-\frac{1}{2}\)	\(-\sqrt{3}\)	\(\frac{2\sqrt{3}}{3}\)	\(-2\)	\(-\frac{\sqrt{3}}{3}\)
\(135\ ^{\circ}\)	\(\frac{3 \pi}{4}\)	\(\frac{\sqrt{2}}{2}\)	\(-\frac{\sqrt{2}}{2}\)	\(-1\)	\(\sqrt{2}\)	\(-\sqrt{2}\)	\(-1\)
\(150\ ^{\circ}\)	\(\frac{5 \pi}{6}\)	\(\frac{1}{2}\)	\(-\frac{\sqrt{3}}{2}\)	\(-\frac{\sqrt{3}}{3}\)	\(2\)	\(-\frac{2\sqrt{3}}{3}\)	\(-\sqrt{3}\)
\(180\ ^{\circ}\)	\(\pi\)	\(0\)	\(-1\)	\(0\)	\(\infty\)	\(-1\)	\(\infty\)

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