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, you can directly measure sine, cosine and tangent.

What happens when the angle, θ is 0°?

cos=1, sin=0 and tan=0

What happens when θ is 90°?

cos=0, sin=1 and tan is undefined

View Larger

Try It!

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

Notice that 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°) = ^√1/₂ = ¹/₂ (because √1 = 1)
sin(45°) = ^√2/₂
sin(60°) = ^√3/₂

And cos goes "3,2,1"

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

What about tan?

tan = sin/cos, so you can calculate:

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

Where did those 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 will be:

(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

Summary

√½ is usually changed to this:
And √¾ is usually changed to this:

And here is the result (as before):

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

Putting it All Together

And here they are for every quadrant. With the correct sign (plus or minus) as per Cartesian Coordinates.

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

Unit Circle Degrees

And this is the same Unit Circle in radians.

Unit Circle Radians