Free maths tool · Level: Ages 14 to 18
Unit circle
Drag the point around the circle and see what sine, cosine and tangent actually are: lengths on a drawing.
On the unit circle (radius 1), the cosine of an angle is the x coordinate of the point and the sine is the y coordinate. Tangent is read off the line x = 1. That is why sine always stays between -1 and 1: the point cannot go further than the circle.
Why sin and cos never go above 1
The circle has radius 1, so the point you drag is always exactly 1 away from the origin. Cosine is its horizontal distance, sine its vertical distance. Neither can be bigger than the radius itself.
That is the fastest check in an exam. If you get sin(x) = 1.4, something went wrong, because the circle does not reach that far.
Where tangent comes from
Extend the radius until it hits the vertical line x = 1. The height where it crosses is the tangent. Hence the name: that line touches the circle, it is a tangent line.
Set the angle to 90 degrees and the radius runs parallel to that line. They never meet, which is why tan(90°) does not exist. Not a rule to memorise, just something you watch happen.
From circle to sine wave
The graph underneath is not separate material. It is the same motion unrolled: angle along the bottom, the height of your point up the side.
Drag once around and follow the two dots on the graph. The sine wave is literally the trail your point leaves behind.
The exact values
At 30°, 45°, 60° and their multiples the point snaps and the exact value appears instead of a decimal. Those are the angles you need without a calculator.
Watch the quadrant readout too. The signs of sin and cos follow from where the point sits, so there is no mnemonic to remember.
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Frequently asked questions
- What is the unit circle?
- A circle of radius 1 centred on the origin. Every point on it corresponds to an angle, and that point’s coordinates are (cos α, sin α). It turns trigonometry into geometry instead of a list of formulas.
- Why is it called the unit circle?
- Because the radius is exactly one unit. That makes the arithmetic simple: the coordinates of the point are the cosine and sine directly, with no dividing by the radius.
- Why does tan(90°) not exist?
- Tangent is read on the line x = 1. At 90 degrees the radius points straight up, parallel to that line, so there is no crossing point. Set the angle just next to 90° and watch the value run away.
- How do I remember the signs in each quadrant?
- You do not need to if you picture the circle. Cosine is the x coordinate, so it is negative on the left. Sine is the y coordinate, so it is negative at the bottom. The tool shows the quadrant as you drag.
- Is this tool free?
- Yes, free and no account needed. If you want someone alongside you while you practise, you can book a 1-on-1 tutor.
More interactive maths tools
- Derivative and tangent →Drag the point and watch the tangent tilt, with the secant closing in on it.
- Integral and Riemann sum →Push the rectangle count up and watch the sum close in on the exact area.
- Normal distribution →Drag the bounds across the bell curve and read probability and z-scores straight off.