Unit Circle

 

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.

 

Sine, Cosine and Tangent

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

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

  • cos 0° = 1, sin 0° = 0 and tan 0° = 0

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

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.

 

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
(the 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 Sin Cos Tan=Sin/Cos
30° 1 √3 = √3 3
45° 1
60° √3

How To Remember?

To help you remember, think "1,2,3" :

  sin(30°)  =   1   =   1   (because √1 = 1)
2 2
  sin(45°)  =   2
2
  sin(60°)  =   3
2

 

And cos goes "3,2,1"

  cos(30°)  =   3
2
  cos(45°)  =   2
2
  cos(60°)  =   1   =   1   (because √1 = 1)
2 2

Just 3 Numbers

In fact, knowing 3 numbers is enough:  1  ,   √2   and   √3
2 2 2

Because they work for cos as well as sin:

 

What about tan?

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

tan(30°) = sin(30°)  =  1/2  =  1
cos(30°) √3/2 √3
  But writing 1/√3 may cost you marks
(see Rational Denominators), so instead use √3/3
tan(45°) = sin(45°)  =  √2/2  = 1 
cos(45°) √2/2
   
tan(60°) = sin(60°)  =  √3/2  = √3 
cos(60°) 1/2
   

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  

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

12 + (√3)2 = 22
1 + 3 = 4
 
Then use sohcahtoa for sin, cos or tan  

Example: sin(30°)

Sine: sohcahtoa

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

 

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°) ?

 

Make a sketch like this, and we can see it is the "long" value:   √3
2

And this is the same Unit Circle in radians.

Unit Circle Radians

Example: What is sin(7π/6) ?

 

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

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

 

 

 

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):

45 Degrees

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

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

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:

√(½) is also this:  
And √(¾) is also this:  

And here is the result (same as before):

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