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Unit Circle

The "Unit Circle" is just 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

 

Try It!

Have a try! Drag the corner around to see how different angles (in radians) 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 vary between positive and negative also.

 

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:

cos2 + sin2 = 1 (a useful "identity")

 

Calculating 30°, 45° and 60°

Let's use that to find the lengths of x and y (which are equal to cos and sin when the radius is 1) for 30°, 45° and 60°:

45 Degrees

For 45 degrees, x and y are equal, so x2 + y2 = 12 becomes 2(x2)=1, so x = √(1/2) = 0.7071...

So, for 45°:

  • cos = √(½) = 0.7071...
  • sin= √(½) = 0.7071...

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 = 12,
becomes: ¼ + y2 = 1,
becomes: y = √(1-¼) = √(¾)

So, for 60°:

  • cos = ½ = 0.5
  • sin = √(¾) = 0.8660...

30 Degrees

And 30° is just 60° swapped over:

  • cos = √(¾) = 0.8660...
  • sin = ½ = 0.5


Pattern

Because ½ = √(¼), there is actually a nice pattern:

Angle Sin Cos Tan=Sin/Cos
30° √(¼) √(¾) 1/√3
45° √(½) √(½) 1
60° √(¾) √(¼) √3

...which should help you to remember them!

Putting it all together

Now, let's show all these angles, for every quadrant. Just make sure we are careful about sign (plus or minus) as per cartesian coordinates:

Square Root of ½ and ¾

I have used √(½) and √(¾) because they are easy to remember, but you may prefer the "simplified" values √(2)/2 and √(3)/2.

Use whichever is easiest for you



Radians

This is the Unit Circle in radians.

You can see we have sin and cos for many simple fractions of pi.


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