MathematicsTrigonometric functionsClass 10Class 11

Unit circle trigonometry: interactive simulation

On a circle of radius 1, the point at angle θ is (cos θ, sin θ). That single picture defines sin and cos for every angle, not just the ones in a right triangle.

Useful for: JEE Main, CBSE Boards

Try this

Drag the point around. The signs of sin, cos and tan change from quadrant to quadrant, and the graphs on the right are just these lengths unrolled against the angle.

  1. 1

    Snap to 30°, 45° and 60°.

    What you should see

    sin goes 1/2, 1/√2, √3/2 while cos goes the opposite way.

  2. 2

    Move from the 1st quadrant to the 2nd.

    What you should see

    cos and tan turn negative, sin stays positive — the “S” in All-Sin-Tan-Cos.

  3. 3

    Move towards 90°.

    What you should see

    cos approaches 0, so tan = sin/cos grows without limit and is undefined at 90°.

  4. 4

    Compare 30° with 150°.

    What you should see

    Same sin (0.5), opposite cos. This is sin(180° − θ) = sin θ.

Formulas behind it

Point on the unit circle

(x,y)=(cos⁡θ,sin⁡θ)(x, y) = (\cos\theta, \sin\theta)

Pythagorean identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

Tangent

tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}

Supplementary angle

sin⁡(180∘−θ)=sin⁡θ\sin(180^\circ - \theta) = \sin\theta

Common mistakes

  • Mixing degrees and radians on a calculator.
  • Using the sign of the angle in the 1st quadrant for every quadrant (forgetting ASTC).
  • Writing tan 90° = ∞ as a value. It is undefined.
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FAQ

Why is the radius taken as 1?

With radius 1, the hypotenuse is 1, so sin θ and cos θ are simply the y and x coordinates of the point.

What does ASTC mean?

All positive in Q1, Sin in Q2, Tan in Q3, Cos in Q4 — the functions that are positive in each quadrant.

What is the period of sin and tan?

sin and cos repeat every 360° (2π); tan repeats every 180° (π).

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