# Unit Circle – Definition, Chart, Equation, Examples, Facts

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## What is a Unit Circle in Math?

A unit circle is a circle of unit radius with center at origin. A circle is a closed geometric figure such that all the points on its boundary are at equal distance from its center. For a unit circle, this distance is 1 unit, or the radius is 1 unit.

Let us learn the equation of the unit circle, and understand the ways to represent each of the points on the circumference of the unit circle, with the help of T-ratios.

## Unit Circle: Definition

The unit circle is a circle with a radius of one unit.

The unit circle is fundamentally related to the concepts of trigonometry. The trigonometric functions can be defined using the unit circle.

A unit circle on the Cartesian plane is shown below. It has its center at origin and all the points on the circumference are at a distance of 1 unit from the center.

## Equation of a Unit Circle

The general equation of a circle is of the form $(x \;-\; a)^2 + (y \;-\; b)^2 = r^2$, where the center of the circle is (a, b) and the radius is r. A unit circle in the x-y plane is formed with a center at origin (0,0) and radius 1.

Thus, the equation $(x \;-\; a)^2 + (y \;-\; b)^2 = r^2$ becomes

$(x \;-\; 0)^2 + (y \;-\; 0)^2 = 1^2$

$x^2 + y^2 = 1$

Thus, the equation of the unit circle on an x-y plane is $x^2 + y^2 = 1$.

## Finding Trigonometric Functions Using a Unit Circle

We can find trigonometric ratios using the unit circle. Consider the right triangle constructed in the unit circle shown in the diagram below.

Let P(x,y) be any point on the circle such that the line joining the origin and the point P makes angle with the positive x-axis. The radius of the circle (length = 1 unit) represents the hypotenuse of the right triangle. So, the sides of the right triangle are 1, x, y.

By the definition of the trigonometric ratios, we have

$sin\; \theta = \frac{Opposite\; Side}{Hypotenuse} = \frac{y}{1}$

$\Rightarrow y = sin\theta$

$cos\; \theta = \frac{Adjacent\; Side}{Hypotenuse} = \frac{x}{1}$

$\Rightarrow x = cos\theta$

Thus, $P(x,y) = P(cos\theta ,\; sin\theta )$

## Unit Circle with Sin, Cos, and Tan

We just learnt that coordinates of any point on the unit circle are equal to $(cos\; \theta,\; sin\; \theta )$.

Thus, $x = cos\; \theta$ and $y = sin\; \theta$

Using these values, we can further calculate $tan\; \theta = \frac{sin\; \theta}{cos\; \theta}$

Example: Find the value of tan $60^\circ$ using sin and cos values from the unit circle.

We know that, tan$\;60^{\circ} = \frac{sin\; 60^\circ}{cos\; 60^\circ}$

Refer to the unit circle chart. We get the values

$sin\; 60^\circ = \frac{1}{2}$

$cos\; 60^\circ = \frac{\sqrt{3}}{2}$

Therefore, tan $60^\circ = \frac{1}{2}\frac{\sqrt{3}}{2}$

tan $60^\circ = \frac{1}{3}$

## Unit Circle and Trigonometric Values

The unit circle can be used to find all the trigonometric values. Let’s see some examples.

When $\theta = 0^\circ$, we have $x = 1$ and $y = 0$.

Thus,

$x = cos\; \theta = cos\; 0^\circ = 1$

and

$y = sin\; \theta = sin\; 0^ \circ = 0$

Similarly, for $= 45^\circ$

$x = cos\; 45^\circ = \frac{1}{\sqrt{2}}$ and $y = sin\; 45^\circ = \frac{1}{\sqrt{2}}$.

If a right triangle is placed in a unit circle in the cartesian coordinate plane, with hypotenuse, base, and altitude measuring 1, x, y units respectively, then the unit circle identities can be given as,

• $sin\;(\theta) = y$
• $cos\;(\theta) = x$
• $tan\;(\\theta) = \frac{sin\; \theta}{cos\; \theta} = \frac{y}{x}$
• $sec\;(\theta) = \frac{1}{x}$
• $cosec\;(\theta) = \frac{1}{y}$
• $cot\;(\theta = \frac{cos\; \theta}{sin\; \theta} = \frac{x}{y}$

In the next section, we will look at the unit circle with radians and unit circle degrees.

## Unit Circle and Pythagorean Identities

Let us observe how we derive these unit circle equations considering a unit circle.

A point on the unit circle can be represented by the coordinates $cos\; \theta$ and $sin\; \theta$ .

## Frequently Asked Questions on Unit Circle

A line that touches the circle at one point is called a tangent to the circle.

The intersecting point at which the tangent touches the circle is known as its point of contact.

Trigonometry is used in various fields like astronomy, oceanography, electronics, navigation, etc.

The radius of a unit circle is 1 unit.

The unit circle gives the values of sine and cosine function. However, the unit circle with tangent gives the values of the tangent function or tan function for different angles between 0 and 360 degrees.