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# Unit Circle

Let's look at the unit circle, how to construct one, and what it is useful for in maths.

## What is the unit circle?

The unit circle has a radius of 1, with a centre at the origin (0,0). Therefore the formula for the unit circle is

This is then used as a basis in trigonometry to find trigonometric functions and derive Pythagorean identities.

The unit circle

We can use this circle to work out the sin, cos and tan values for an angle 𝜃 between 0 ° and 360 ° or 0 and 2𝜋 radians.

Sin, cos and tan on the unit circle

## What is the unit circle used for?

For any point on the circumference of the unit circle, the x-coordinate will be its cos value, and the y-coordinate will be the sin value. Therefore, the unit circle can help us find the values of the trigonometric functions sin, cos and tan for certain points. We can draw the unit circle for commonly used angles to find out their sin and cos values.

The unit circle Image: public domain

The unit circle has four quadrants: the four regions (top right, top left, bottom right, bottom left) in the circle. As you can see, each quadrant has the same sin and cos values, only with the signs changed.

## How to derive sine and cosine from the unit circle

Let's look at how this is derived. We know that when 𝜃 = 0 ° , sin𝜃 = 0 and cos𝜃 = 1. In our unit circle, an angle of 0 would look like a straight horizontal line:

The unit circle for 𝜃 = 0

Therefore, as sin𝜃 = 0 and cos𝜃 = 1, the x-axis has to correspond to cos𝜃 and the y-axis to sin𝜃. We can verify this for another value. Let's look at 𝜃 = 90 ° or 𝜋 / 2.

The unit circle for 𝜃 = 90

In this case, we have a straight vertical line in the circle. We know that for 𝜃 = 90 ° , sin 𝜃 = 1 and cos 𝜃 = 0. This corresponds to what we found earlier: sin 𝜃 is on the y-axis, and cos 𝜃 is on the x-axis. We can also find tan 𝜃 on the unit circle. The value of tan 𝜃 corresponds to the length of the line that goes from the point on the circumference to the x-axis. Also, remember that tan𝜃 = sin𝜃 / cos𝜃.

The unit circle for sin, cos and tan

## The unit circle and Pythagorean identity

From Pythagoras' theorem, we know that for a right-angled triangle . If we were to construct a right angled triangle in a unit circle, it would look like this:

The unit circle with sin and cos

So a and b are sin𝜃, and cos𝜃 and c is 1. Therefore we can say: which is the first Pythagorean identity.

## Unit Circle - Key takeaways

• The unit circle has a radius of 1 and a centre at the origin.

• The formula for the unit circle is .

• The unit circle can be used to find sin and cos values for angles between 0 ° and 360 ° or 0 and 2𝜋 radians.

• The x-coordinate of points on the circumference of the unit circle represents the cos value of that angle, and the y-coordinate is the sin value.

A unit circle is a circle with a radius of 1 and a centre at the origin used to find values of and understand trigonometric functions like sin, cos and tan for different angles.

Cos is the x-coordinate of a point on the circumference of the circle and sin is its y-coordinate.

The unit circle is used for finding the values of different trigonometric functions for angles in degrees or radians.

## Final Unit Circle Quiz

Question

What is the unit circle?

A unit circle is a circle with a radius of 1 and a centre at the origin used to find values of and understand trigonometric functions like sin, cos and tan for different angles

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Question

The unit circle allows us to look at angles between which two values?

0 and 360° or 0 and 2𝜋 radians

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Question

What is sin on the unit circle?

The y-coordinate of a point on the circle

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Question

What is cos on the unit circle?

The x-coordinate of a point on the circle

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Question

What is tan on the unit circle?

The length of the tangent line that goes from the point on the circumference to the x-axis

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Question

What is the radius of the unit circle?

1

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Question

What is the centre of the unit circle?

The origin (0,0)

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Question

How many quadrants are in a circle?

Four

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Question

What is the sin value of 90°?

1

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Question

What is the cosine value of 0°?

1

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Question

What is the cos of 𝜋/3?

1/2

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Question

What is the sin of 7𝜋/6?

-1/2

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Question

What is the sin of 𝜋?

0

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