Unit Circle


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What is a Unit Circle?

A unit circle means a circle with radius 1 unit.

A unit circle in math is an important topic which helps us understand the trigonometric functions in the cartesian coordinate plane.

Let us see how to use the Pythagoras theorem and the unit circle to understand the trigonometric functions of sine, cosine and tangent.


Formula for Unit Circle

Euation of a circle in the cartesean coordinate plane

The equation of a circle is given by the formula:

\((x-h)^2 +(y-k)^2 = r^2\)

where \((h,k)\) are the coordinates of the centre in the cartesian plane and r is the radius of the circle.
(\(x,y\)) are the points on the circumference of the circle which are at a distance \(r\) (radius) from the center(\(h,k\)).

 

In case of a unit circle, the centre lies at (0,0) and the radius is 1 unit.

Unit circle math - the centre lies at (0,0) and the radius is 1 unit.

Thus, the formula for the unit circle is:

\( x^2 +y^2 = 1\)

 Calculating Trigonometric Functions Using Unit Circle

We calculate the trigonometric functions sine, cosine and tangent using unit circle.

Let us apply the Pythagoras theorem in a unit circle to understand the trigonometric functions.

Consider a right triangle placed in a unit circle in the cartesian coordinate plane.

The trigonometric functions are shown below.

Unit circle trigonometry - the centre lies at (0,0) and the radius is 1 unit. We will apply the Pythagoras theorem to get the values of sin and cos.

Cosine is the x-coordinate and Sine is the y-coordinate.

Example when \(\theta = 0 \)

Sin 0 in unit circle

x-coordinate is 1 and y-coordinate is 0

Therefore,

  • \(\cos0^\circ = 1\)
  • \(\sin 0^\circ = 0\)

Can you identify the value of \(\cos 90^\circ\) and \(\sin 90^\circ \)

cos 90 and sin 90 value from the unit circle graph

Let us use this unit circle and find the important trigonometric function values when 

\( \theta \text{ is}\:  30^\circ, 45^\circ, 60^\circ \)

Note that we can also measure in radians

We know that 360° = 2 \(\pi\) Radians.

We can convert the angular measures to radian measures and express in terms of the radians.

Table- 1

Angle \(\theta\) Radians Sin Cos Tan =\(\begin{align}\frac{\sin\theta}{\cos\theta} \end{align}\)
 0° 0 0 1 0
30° \(\begin{align}\frac{\pi}{6}\end{align}\) \(\begin{align}\frac{1}{2}\end{align}\) \(\begin{align}\frac{\sqrt{3}}{2}\end{align}\) \(\begin{align}\frac{\sqrt{3}}{3}\end{align}\)
45° \(\begin{align}\frac{\pi}{4}\end{align}\) \(\begin{align}\frac{\sqrt{2}}{2}\end{align}\) \(\begin{align}\frac{\sqrt{2}}{2}\end{align}\) 1
60° \(\begin{align}\frac{\pi}{3}\end{align}\) \(\begin{align}\frac{\sqrt{3}}{2}\end{align}\) \(\begin{align}\frac{1}{2}\end{align}\) \(\sqrt{3}\)
90° \(\begin{align}\frac{\pi}{2}\end{align}\) 1 0 undefined

We can find the Secant, Cosecant and Cotangent functions using these formulas:

Secant (\(\sec\theta\) )= \(\begin{align}\frac{1}{\sin \theta}\end{align}\)
Cosecant (\(\csc\theta\) ) = \(\begin{align}\frac{1}{ \cos\theta}\end{align}\)
Cotangent (\( \cot\theta\) )= \(\begin{align}\frac{1}{\tan\theta}\end{align}\)

We have discussed the unit circle for the first quadrant.

Similarly, we can extend and find the radians for all the unit circle quadrants.

The numbers \(\begin{align} \frac{1}{2} , \frac{\sqrt 2}{2}, \frac{\sqrt3}{2}, 0\: \text{and}\:1 \end{align} \) repeat along with the sign in all 4 quadrants.

Cartisian coordinates in the four quadrants

Visit the unit circle simulation and verify the values at all four quadrants. 

Unit circle chart in Angular degrees

Unit circle trignometry angles and radians

Unit circle chart in Radians

Unit circle radians

Unit circle Simulation

Select a Quadrant and drag the point P in the simulation below to visualise the unit circle in all four quadrants.

 
tips and tricks
Tips and Tricks 
1. All points on the unit circle alter between \(\begin{align}\frac{1}{2} , \frac{\sqrt 2}{2}, \frac{\sqrt3}{2}, \: 0 \:\text{and}\: 1 \end{align} \) (both positive and negative)
  Vertical lines indicate the sin value and horizontal lines indicate the cos value.
  Unit circle - tips and tricks to remember the trigonometric values

Unit Circle Calculator

Use the Unit circle calculator below to calculates the radians, sin, cosine and tan values.

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Solved Examples on Unit Circle Formula

Example 1

 

 

Does the point P \(\begin{align}\left(\frac{1}{2}, \frac{1}{2} \right) \end{align}\) lie on the unit circle ?

Solution:

We know that equation of a unit cicle is 

\(x^2 +y^2 = 1\)

Subsutituting \(\begin{align}x = \frac{1}{2}\end{align}\) and \(\begin{align}y = \frac{1}{2} \end{align}\), we get:

\[ \begin{align}&= x^2 +y^2  \\\\
&=  \left(\frac{1}{2}^2 + \frac{1}{2}^2 \right) \\\\
&= \frac{1}{4} +\frac{1}{4} \\\\
&= \frac{1}{2}\\\\
&\neq 0
\end{align}\]

Since \(x^2 +y^2 \neq 1\), the point P \(\begin{align}\left(\frac{1}{2}, \frac{1}{2} \right)\end{align} \) does not lie on the unit circle.

\(\therefore\) P \(\begin{align}\left(\frac{1}{2}, \frac{1}{2}\right)\end{align}\) ) doesn't lie on the unit circle.
Example 2

 

 

Why is \(\cot 180\)° undefined?

Solution:

We know that 

  • \(\begin{align}\cot \theta = \frac{1}{\tan\theta}\end{align}\)
  • \(\begin{align}\tan \theta = \dfrac{\sin \theta}{\cos\theta}\end{align}\)

Therefore

\(\cot \theta = \dfrac{\cos \theta}{\sin \theta}\)

From the unit circle chart, we know that 

\(\sin  180^\circ = 0 \)

Since division by 0 is \(\infty \), \(\cot 180\)° = \(\infty \) 

\( \therefore \cot 180\)° = \(\infty \)   
Example 3

 

 

Use the unit circle chart and identify the following for the Point P\(\begin{align}\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \end{align}\)

a) Angle \(\theta\)
b) Quadrant
c) sin \(\theta\)
d) cos \(\theta\)

Solution:

Unit circle trigonometry angles and radians

Using the unit circle chart, we can say that:

P \(\begin{align}\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) \end{align}\) lies in the III quadrant.

The angle is 225°

  • \( \sin\) 225° is the y -coordinate \(\begin{align} -\frac{\sqrt{2}}{2}\end{align}\)
  • \(\cos\) 225° is the x-coordinate \( \begin{align}-\frac{\sqrt{2}}{2}\end{align}\)
\(\therefore\) The angle is  225° and it lies in the III quadrant.
Example 4

 

 

Find the exact value of \( \tan\) 210° using the unit circle.

Solution:

We know that

\( \tan 210°\) = \(\begin{align}\frac{ \sin 210°}{ \cos210°}\end{align}\)

Using the unit circle chart:

  • \( \sin\) 210° =  -\(\begin{align}\frac{1}{2}\end{align}\)
  • \( \cos\) 210° = -\(\begin{align}\frac{\sqrt{3}}{2}\end{align}\)

Therefore,

 \(\begin{align} \text{tan } 210° &= \frac{ \sin 210°}{\cos 210°} \\\\
&=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} \\\\
&=\frac{1}{\sqrt{3}} \\\\
&=\frac{\sqrt3}{3}
\end{align} \)

\(\therefore \tan\) 210° = \(\begin{align}\frac{\sqrt{3}}{3}\end{align}\)
Example 5

 

 

Find \(\sin\) 900°

Solution:

The unit circle has  0°- 360°

Let us represent 900° in terms of 360° 

900° is 2 full rotations of 360°  and an additional rotation of 180°

 sin 900 in unit circle

Hence, 900° will have the same trigonometric ratio as 180° 

Therefore,

\(\sin\) 900° = \(\sin\) 180°

From the unit circle chart, we know that:

\(\sin\) 180° =  0

\(\therefore \sin\) 900° = 0
 
important notes to remember
Important Notes
  1. Unit circle is a circle of radius 1 unit.
  2. The equation of a unit circle is \( x^2 +y^2 = 1\)
  3. Refer Table- 1 for important Sin Cos and Tan values of the 1st quadrant.

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Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
 
Challenge your math skills
Challenging Questions
  1.  Using the unit circle, prove the identity \(\sin^2 \theta +\cos^2 \theta = 1 \)
  2.  What are the sine, cosine and tangent for the x and y axes (recollect the angle formed at the axes)?

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions(FAQs) 

1. What is a unit circle in math?

A unit circle is a circle of radius 1 unit.

2. What is the unit circle used for?

We use the unit circle to calculate the cosine, sine, and tangent of any angle between 0° and 360° ( 0 and 2π radians).

3. How does unit circle work?

A unit circle in the cartesian coordinate has (0,0) as it's center and radius 1 unit.

This creates a right triangle.

We can then apply the Pythagoras theorem to find the values of sin, cos and tan in the unit circle quadrants.

 

Unit circle trignometry

4. How do you solve a unit circle problem?

Use the unit circle chart. 

The point P gives the coordinates on the unit circle.

Use it to find the trigonometric function values.

For example: To find sin 225° and cos 225°, use the unit circle simulation above.

Fix θ as 225° and locate the coordinates.

x-coordinate gives the cosine and y-coordinate gives the sin values.

We can see that:

  • sin 225° = -0.71
  • cos 225° = -0.71

Unit circle sin and cosine 225 degree from the unit circle chart

  
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