# Unit Circle

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Unit Circle is an important and easy to learn concept which is required to understand periodicity, trigonometric function, graphs and many more things.

#### Contents

## What exactly is a Unit Circle?

As the name suggests it is a circle with radius equals to \(1\) unit, drawn on Cartesian plane with its center placed at the origin.

## What is the need of a Unit Circle?

\[ \begin{align} \sin (\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c}\\ \cos (\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c}\\ \tan (\theta) &= \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a}\\ \end{align}\]

\[ \begin{array} { | c | c | c | } \hline \text{angle measure, } \theta & \sin \theta & \cos \theta\\ \hline 0 & 0 & 1 \\ \hline \dfrac{\pi}{6} & \dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \\ \hline \dfrac{\pi}{4} & \dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{2}}{2} \\ \hline \dfrac{\pi}{3} & \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ \hline \dfrac{\pi}{2} & 1 & 0 \\ \hline \end{array} \]

The trigonometry we are familiar with so far is based on only right triangles and acute angles. However, with help of Unit Circle we can extend our understanding of trigonometric functions plus also become familiar with the use of non-acute angles.

But before starting with Unit Circle you must be familiar with the circular system of angle measurement .

## Getting started with a Unit Circle

Let us understand a few properties of a Unit Circle,

A right triangle \(\bigtriangleup AOB\) right angled at \(A\), lie on the Cartesian plane such that \(OA\) lies on the \(x-axis\) , point \(O\) lies on the origin and point \(B\) lies anywhere on the Unit Circle. Note that \(OB=1\) units.

Now,

\[ \begin{align} \sin (\theta) &= \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AB}{OB} = \frac{AB}{1} = AB\\ \cos (\theta) &= \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{OA}{OB} = \frac{OA}{1} = OA\\ \end{align}\]

So,

In that case what can we say about the coordinates of point \(B\)? Recall that it is \(cos (\theta)\) units away from the \(y-axis\) and \(\sin (\theta)\) units away from the \(x-axis\). Clearly the \(x-coordinate\) will be \(\cos (\theta)\) and \(y-coordinate\) will be \(\sin (\theta)\).

This result is common for all the values of \(\theta\).

This also tells us that if we to figure out \(sin\) or \(\cos\) of any angle \(\theta\) then we just to look at the coordinates of point on unit circle.

## How to measure an angle?

An angle on Unit Circle is always measured from the \(x-axis\).

An angle is said to be positive if it is measured by going in anticlockwise direction from the \(x-axis\) and negative if it is measured by going in clockwise direction from the \(x-axis\).

## Example Problems

Given that a line passes through the unit circle with the angle \(\theta=\dfrac{\pi}{4}\) find the x,y. Refer to the given diagram.

We know that \(\begin{cases} \cos(\theta)= x \\ \sin(\theta)=y\end{cases}\). put the value of theta to find\[\begin{cases} x=\dfrac{\sqrt{2}}{2}\\y=\dfrac{\sqrt{2}}{2}\end{cases}\]We are done\(_\square\)

A line passes through the unit circle at the point at \(x=\dfrac{1}{2}\). find the value of \(\tan^2(\theta)\)

first, we know that \(x^2=\cos^2(\theta)=\dfrac{1}{4}\). we also know \(y^2=\sin^2(\theta)=1-cos^2(\theta)=\dfrac{3}{4}\). using another trig identity we have \(\tan^2(\theta)=\dfrac{\sin^2(\theta)}{\cos^2(\theta)}=\boxed{3}\) we are done\(_\square\)

**Cite as:**Unit Circle.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/unit-circle-basic-concept-for-higher-trigonometry/