Unit Circle

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 = r², 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 = 1². This is explained to obtain the equation of a unit circle.

Equation of a Unit Circle: x² + y² = 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