Unit Circle

The unit circle is considered a circle with radius 1. It is very simple and so there are great ways to learn it and talk about its angles and length. The center is to put on a graph where the x-axis and y-axis just cross and then we get a neat diagram of this unit circle.

unit circle

Properties of Unit Circle

A point on the unit circle is called a terminal point, which consist of the x and y coordinate:

x = cos A
y = sin A
y/x = tan A

Due to the radius = 1, we can measure sine, cosine, and tangent easily. The answers to questions that arise in our minds, are given below –

1. What happens when the angle is 0°?

So, from the graph of sine, cosine, and tangent and from this unit circle, we can say that:

Sin 0° = 0
Cos 0° = 1
Tan 0° = 0/1 = 0

2. What happens when the angle is 90°?

According to the graph of sine, cosine, and tangent, we can say that:

Sin 90° = 1
Cos 90° = 0
Tan 90° = 1/0 = not defined

3. What happens when the angle is 180°?

Sin 180° = 0
Cos 180° = -1
Tan 180° = 0/-1 = 0

4. What happens when the angle is 270°?

Sin 270° = -1
Cos 270° = 0
Tan 270° = -1/0 = not defined

5. What happens when the angle is 360°?

360 degrees brings the unit circle back to the beginning (which is 0°).

Sin 90° = 1
Cos 90° = 0
Tan 90° = 1/0 = not defined

You can also try out the interactive unit circle.

First, the Pythagoras theorem shows that in any right-angled triangle, the square of the long side is equal to the square of the sum of other both sides:

X2 + y2 = 12
But the square of 1 is only 1, so:
X2 + y2 = 12 (equation of unit circle)

also, x = cos and y =sin; we will get –

(cos (θ)) + (sin (θ)) = 1 (identity)

You must remember the value of sin, cos and tangent for common the angles of 30, 45 and 60. It is difficult to remember things but it will make your work, calculation, and life easier when you know it, not only in exams but every time you need to estimate quickly.

There is given below values of the table.

table

cartesian

Now, let us see the Cartesian coordinates of the unit circle. After crossing the x and y-axis, they are divided into 4 quadrants which are 1st, 2nd, 3rd, and 4th. The first quadrants all have positive values.

The second quadrant has the sin negative with cosec and tangent and cot. The cos-value is positive and sec value is also positive. The third quadrant has both negative, which means all values are negative, except tangent and cot.

The fourth quadrant has cos negative value with all the negative values except sine and cosec.

The below diagram shows you the cartesian of a whole circle with the easy to learn method. The first coordinate is of cos and the second is of sine.

coordinate

Example 1 – find cos (5π/6).

Using the unit circle from above, locate the angle 5π/6 (or 150°, if converted to degrees).

The terminal point is (-√3/2, ½).

The cos of 5π/6 would be the x-coordinate, or -√3/2, so the answer is -√3/2.

Example 2 – find sin (π/4).

Using the unit circle from above, locate the angle π/4 (or 45°, if converted to degrees).

The terminal point is (√3/2, ½).

The sin of π/4 would be the y-coordinate, or ½, so the answer is ½.

Example 3 – find tan (π/2).

Locate the angle. If converted to degrees, it is 90°, which is the very top of the unit circle.

The terminal point is (0,1).

The tan of 90° is x/y, or 0/1 = 0. The answer is zero.