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.
Point P of course has coordinates.
We write these as (xP, yP).
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 =
We looked at an example in which 0° < α < 90°.
For α ≤ 0 or α ≥ 90° these formulas also apply.
See the examples below.
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(–0.84°) ≈ 147°
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.
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.
sin–1(0.91°) ≈ 66°
sin–1(0.91°) ≈ 66° (see example above)
α ≈ 180° – 66° = 114°