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Unit Circle


Unit circle is a circle with a radius of one and is denoted by 'S$^{1}$', and is the generalization to higher dimensions in the unit sphere. Unit circle is centered at the origin in trigonometry. As the radius is one, you can directly measure sine, cosine and tangent. Equation of unit circle is x$^{2}$ + y$^{2}$ = 1. If a point on the unit circle is (x, y), then |x| and |y| are the lengths of the legs of a right triangle, whose hypotenuse has length 1. Plot the tangent and then find the equation of the unit circle by using pythagoras theorem.
For the right angle triangle, square of the longest side must be equal to the square of the rest of two sides. Hence, this will give us the equation of unit circle. By Pythagorean theorem, x and y satisfies the equation x$^{2}$ + y$^{2}$ = 1.

Unit Circle Chart

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Given below is the chart of the unit circle:

Sl. No
$\theta$ (rad) $\theta^{0}$
Sin $\theta$ Cos $\theta$
1 $\frac{\pi}{6}$ 30 $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{3}}$
2 $\frac{\pi}{3}$ 60 $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$
3 $\frac{\pi}{2}$ 90 1 0 $\infty$
4 $\frac{2\pi}{3}$ 120 $\frac{\sqrt{3}}{2}$ - $\frac{1}{2}$ -$\sqrt{3}$
5 $\frac{5\pi}{6}$ 150 $\frac{1}{2}$ - $\frac{\sqrt{3}}{2}$ -$\frac{1}{\sqrt{3}}$
6 $\pi$ 180
0 -1 0
7 $\frac{7\pi}{6}$ 210 - $\frac{1}{2}$ -$\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{3}}$
8 $\frac{4\pi}{3}$ 240 - $\frac{\sqrt{3}}{2}$ -$\frac{1}{2}$ $\sqrt{3}$

Use the below widget to calculate unit circle value.

Trigonometry Unit Circle

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In trigonometry, the unit circle is centered at the origin. We can examine angle $\theta$ in the unit circle as follows:
cos$\theta$ = $\frac{\text{Adjacent}}{\text{Hypotenuse}}$ = $\frac{x}{1}$

Sin $\theta$ = $\frac{\text{Opposite}}{\text{Hypotenuse}}$ = $\frac{y}{1}$

$\Rightarrow$ cos $\theta$ = x and sin $\theta$ = y
Therefore, (x, y) = (Cos $\theta$, Sin$\theta$)

Sine and Cosine are represented by vertical leg and horizontal leg respectively.

As x$^{2}$ + y$^{2}$ = 1, we have cos$^{2}$ $\theta$ + Sin$^{2}$ $\theta$ = 1. The graph of the unit circle is shown below:

Trigonometry Unit Circle
Given below are some of the important angles in trigonometry and are summarized in the figure below:

Look at the point corresponding to point 'a', say a = 30$^{\circ}$ ($\frac{\pi}{6}$).

The x - coordinate is $\frac{\sqrt{3}}{2}$ and the y-coordinate is $\frac{1}{2}$.

Therefore, cos $\frac{\pi}{6}$ = $\frac{\sqrt{3}}{2}$ and sin $\frac{\pi}{6}$ = $\frac{1}{2}$.

Unit Circle