Geometry

Unit circle

Turn a point. Trace sine, cosine and tangent.

Predict, test and explain

Step 1 of 3

Predict

At 90° on the unit circle, what are the point coordinates?

θ=45∘\theta = 45^\circ
P≈(0.71,  0.71)P \approx (0.71,\;0.71)
unit circle\text{unit circle}
−1-1
00
11
0∘0^\circ
90∘90^\circ
180∘180^\circ
270∘270^\circ
360∘360^\circ
sin⁡θ\sin\theta

Scroll the diagram sideways to see every label.

sin⁡θ≈0.707\sin\theta \approx 0.707cos⁡θ≈0.707\cos\theta \approx 0.707tan⁡θ=1\tan\theta = 1

InvestigateWhere do sine and cosine swap values?

Explore and compare
Can you break the rule?

Different angles in one full turn have different sine values.

Always, sometimes or never? Test it in the simulation.

Hint

Look for two circle points at the same height, then two at different heights.

Check the reasoning

Sometimes. Sine gives the vertical coordinate. Distinct points can have equal heights, but they need not.

sin⁡30∘=sin⁡150∘=12\sin30^\circ=\sin150^\circ=\tfrac12

One counterexample disproves “always”. Examples alone do not prove it.

Find two different angles in one turn with the same sine.

Try it in the simulation. Change one thing at a time.

Hint

Look for two points at the same height on the circle. Select sine.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

On the unit circle, an angle is measured anticlockwise from the positive x-axis. Find the point's exact coordinates at 120 degrees.

Hint

Use cosine for x and sine for y. The point is in the second quadrant.

Reveal answer

The x-coordinate is negative and the y-coordinate is positive.

(x,y)=(cos⁡120∘,sin⁡120∘)=(−12,32)(x,y)=(\cos120^\circ,\sin120^\circ)=\left(-\frac12,\frac{\sqrt3}{2}\right)

Build an explanation

Hint 1

Follow the point's horizontal and vertical positions.

Hint 2

On a circle of radius 1, the coordinates are (cos θ, sin θ).

Hint 3

Tangent is sin θ divided by cos θ; it is undefined when cos θ is zero.

Worked example

This example uses fixed values, separate from the diagram controls.

Find sine, cosine and tangent at 45°.

  1. The horizontal and vertical coordinates are equal at 45°.
  2. Each coordinate is √2/2.
  3. Divide sine by cosine to find tangent.

Answer: sin 45° = cos 45° = √2/2; tan 45° = 1.

Connect this idea

Watch out: Tangent at 90° is undefined, rather than a very large finite value.