Unit Circle

unit circle center at (0,0)

 

The "Unit Circle" is a circle with a radius of 1.

Being so simple, it is a great way to learn and talk about lengths and angles.

The center is put on a graph where the x axis and y axis cross, so we get this neat arrangement here.

 

unit circle center at (0,0)

Sine, Cosine and Tangent

Because the radius is 1, we can directly measure sine, cosine and tangent.

unit circle center angle 0

What happens when the angle, θ, is 0°?

cos 0° = 1, sin 0° = 0 and tan 0° = 0

unit circle center angle 90

What happens when θ is 90°?

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

Try It Yourself!

Have a try! Move the mouse around to see how different angles (in radians or degrees) affect sine, cosine and tangent

../algebra/images/circle-triangle.js

The "sides" can be positive or negative according to the rules of Cartesian coordinates. This makes the sine, cosine and tangent change between positive and negative values also.

 

Also try the Interactive Unit Circle.

 

unit circle center at (0,0)

Pythagoras

Pythagoras' Theorem says that for a right angled triangle, the square of the long side equals the sum of the squares of the other two sides:

x2 + y2 = 12

But 12 is just 1, so:

x2 + y2 = 1
  equation of the unit circle

Also, since x=cos and y=sin, we get:

(cos(θ))2 + (sin(θ))2 = 1
a useful "identity"

Important Angles: 30°, 45° and 60°

You should try to remember sin, cos and tan for the angles 30°, 45° and 60°.

Yes, yes, it is a pain to have to remember things, but it will make life easier when you know them, not just in exams, but other times when you need to do quick estimates, etc.

These are the values you should remember!

Angle Cos Sin Tan=Sin/Cos
30° √32 12 1 √3 = √3 3
45° √22 √22 1
60° 12 √32 √3

How To Remember?

unit circle 123

To help you remember, cos goes "3,2,1"

 cos(30°)  =  32

 cos(45°)  =  22

 cos(60°)  =  12  =  12

 

And, sin goes "1,2,3" :

 sin(30°)  =  12  =  12  (because √1 = 1)

 sin(45°)  =  22

 sin(60°)  =  32

Just 3 Numbers

In fact, knowing 3 numbers is enough: 12 ,  √22  and  √32

Because they work for both cos and sin:

unit circle cos 1/2, root2/2, root3/2       unit circle cos 1/2, root2/2, root3/2

Your hand can help you remember:

unit circle cos 1/2, root2/2, root3/2       

For example there are 3 fingers above 30°, so cos(30°) = 32

What about tan?

Well, tan = sin/cos, so we can calculate it like this:

tan(30°) =sin(30°)cos(30°) = 1/2√3/2 = 1√3 = √33 *

tan(45°) =sin(45°)cos(45°) = √2/2√2/2 =

tan(60°) =sin(60°)cos(60°) = √3/21/2 = √3 

* Note: writing 1√3 may cost you marks so use √33 instead (see Rational Denominators to learn more).

Quick Sketch

Another way to help you remember 30° and 60° is to make a quick sketch:

Draw a triangle with side lengths of 2   triangle 60 60 with sides of 2

Cut in half. Pythagoras says the new side is √3

a2 + b2 = c2
12 + (√3)2 = 22
1 + 3 = 4  
  triangle 30 60 with sides of 1, 2, root3
Then use sohcahtoa for sin, cos or tan   triangle 30 60 with sides of 1, 2, root3

Example: sin(30°)

Sine: sohcahtoa

sine is opposite divided by hypotenuse
sin(30°) = opposite hypotenuse = 1 2

 

quadrants (+,+) (-,+) (-,-) and (+,-) going counterclockwise

The Whole Circle

For the whole circle we need values in every quadrant, with the correct plus or minus sign as per Cartesian Coordinates:

 

Note that cos is first and sin is second, so it goes (cos, sin):

Unit Circle Degrees

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Example: What is cos(330°) ?

unit circle 330

 

Make a sketch like this, and we can see it is the "long" value:  √32

And this is the same Unit Circle in radians.

Unit Circle Radians

Example: What is sin(7π/6) ?

unit circle 7pi/6

 

Think "7π/6 = π + π/6", then make a sketch.

We can then see it is negative and is the "short" value: −½

 

7708, 7709, 7710, 7711, 8903, 8904, 8906, 8907, 8905, 8908

 

Footnote: where do the values come from?

We can use the equation x2 + y2 = 1 to find the lengths of x and y (which are equal to cos and sin when the radius is 1):

triangle 45 inside unit circle

45 Degrees

For 45 degrees, x and y are equal, so y=x:

x2 + x2 = 1
2x2 = 1
x2 = ½
x = y = √(½)

triangle 30 60 inside unit circle

60 Degrees

Take an equilateral triangle (all sides are equal and all angles are 60°) and split it down the middle.

The "x" side is now ½,

And the "y" side is:

(½)2 + y2 = 1
¼ + y2 = 1
y2 = 1-¼ = ¾
y = √(¾)

30 Degrees

30° is just 60° with x and y swapped, so x = √(¾) and y = ½

And:

1/2 = 2/4 = 24 = 22

Also:

3/4 = 34 = 32

And here is the result (same as before):

Angle Cos Sin Tan=Sin/Cos
30° √32 12 1 √3 = √3 3
45° √22 √22 1
60° 12 √32 √3