## The unit circle definition of sine, cosine, & tangent

## What is Unit Circle?

**A unit circle intends a circle with a radius of 1 unit.**

It is an excellent idea to understand and communicate regarding lengths and trigonometric functions.

## Equation of a Unit Circle

The usual equation of a circle is (x – a)^{2} + (y – b)^{2} = r^{2}, which describes a circle having the center (a, b) and the radius r. This equation of a circle is interpreted to represent the equation of a unit circle. A unit circle is formed with its center at the point(0, 0), which is the beginning of the coordinate axes. and a radius of 1 unit. So the equation of the unit circle is (x – 0)^{2} + (y – 0)^{2} = 1^{2}. This is explained to obtain the equation of a unit circle.

#### Equation of a Unit Circle: x^{2} + y^{2} = 1

Here is the given unit circle, the center lies at (0,0) and the radius is 1 unit. The above equation provides all the points lying on the circle across the four quadrants.

In the fig. where the x-axis and y-axis cross, so we can calculate the neat arrangement here.

## Unit Circle with Sin Cos and Tan

### The trig functions & right triangle trig ratios

As we know the radius of the circle is 1, so we can immediately calculate sine, cosine, and tangent.

**What happens when we put the angle, θ, is 0°?**

- cos 0° = 1,
- sin 0° = 0
- and tan 0° = 0

**What happens when we put the angle, θ, is 90°?**

- cos 90° = 0,
- sin 90° = 1
- and tan 90° = undefined

**Example: Calculate the value of tan 60º using sin and cos values from the unit circle.**

**Solution**:

We know that tan 60° = sin 60°/cos 60°

Using the unit circle chart:

sin 60° = 1/√2

cos 60° = 1/√2

Therefore, tan 60° = sin 60°/cos 60°

= (√3/2)/(1/2)

= √3

Answer: Therefore, tan 60° = √3