# Trigonometry » Unit circle

In the text below all examples are in degrees.
Of course radians can also be used.

## What is a unit circle?

The unit circle is a circle with centre O(0, 0) and a radius of 1. You can draw a point P on the unit circle. The angle of line OP with the horizontal axis, is called α (alpha). Some books use θ (theta). Here, the horizontal axis is always the positive part, so from the origin to (1, 0). If you consider P a moving point, you can also call the angle the rotation angle. This angle always starts at the horizontal axis and goes anti-clockwise. Is your angle going clockwise? Then your rotation angle is negative. So you are rotating backwards.  ### Coordinates

Point P of course has coordinates.
We write these as (xPyP).
You can see this in the following example, where 0° < α < 90°. ## How can you calculate in a unit circle?

Look at the example above. You can recognise a right-angled triangle and within this right-angled triangle you can use sine, cosine and tangent. Because of the radius of the circle, the hypotenuse is always 1. Using this, you will get the following calculations:
`cos`(α) = adjacenthypotenuse = xPOP = xP1 = xP
`sin`(α) = oppositehypotenuse = yPOP = yP1 = yP
`tan`(α) = oppositeadjacent = yPxP
`tan`(α) = oppositeadjacent = `sin`(α)`cos`(α)

Therefore:
xP = `cos`(α)
yP = `sin`(α)

We looked at an example in which 0° < α < 90°.
For α ≤ 0 or α ≥ 90° these formulas also apply.
See the examples below.

 Example 1 xP = `cos`(60°) = 0.5 yP = `sin`(60°) ≈ 0.866 Example 2 xP = `cos`(126°) ≈ –0.588 yP = `sin`(126°) ≈ 0.809 Example 3 xP = `cos`(–78°) = 0.208 yP = `sin`(–78°) ≈ –0.978 Example 4 xP = `cos`(579°) ≈ –0.777 yP = `sin`(579°) ≈ –0.629

In example 4 you can see that when α > 360° the formulas work as well.
As 579° – 360° = 219° you could also do `cos`(219°) ≈ –0.777 here.

### Calculating the angle

When α has to be calculated and xP or yP is known you can use the inverse cosine and sine to calculate the angle. On your calculator you have to use `cos`–1 and `sin`–1.

Example 5 α = `cos`–1(–0.84°) ≈ 147°

However, `cos`–1 always gives an outcome between 0° and 180°. In the next example xP is also –0.84 although we have different rotation angle. Use symmetry to calculate the angle.

Example 6 `cos`–1(–0.84°) ≈ 147° (see example above)
α ≈ 360° – 147° = 213°

yP can also be given. Then you have to use `sin`–1. This will give a result between –90° and 90°. Do you need a different angle? Calculate the angle like in the example above, using symmetry.

Example 7 α = `sin`–1(0.91°) ≈ 66°

Example 8 `sin`–1(0.91°) ≈ 66° (see example above)
α ≈ 180° – 66° = 114°

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