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The Unit CircleFoundation

All six ratios from one point — interactive Mathematics simulation for IIT-JEE.

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Concept

Walk angle θ anticlockwise around a circle of radius 1: the landing point is (cos θ, sin θ) — cosine is the x-shadow, sine the y-shadow. Every identity, sign rule and periodicity is visible: ASTC quadrant signs, sin² + cos² = 1 (it's just Pythagoras), and tan θ as the slope of the radius.

Key formula

P(θ)=(cos⁡θ,sin⁡θ),sin⁡2θ+cos⁡2θ=1,tan⁡θ=sin⁡θcos⁡θP(\theta) = (\cos\theta, \sin\theta), \qquad \sin^2\theta + \cos^2\theta = 1, \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

Derivation

Radius 1 ⇒ the right triangle with legs |cos θ|, |sin θ| has hypotenuse 1 — Pythagoras gives THE identity.

ASTC ('All Students Take Coffee'): Q1 all positive, Q2 sin, Q3 tan, Q4 cos. Any angle reduces to its acute REFERENCE angle: same magnitudes, quadrant sets the signs. Periodicity is one full lap: 2π.

Scenarios to explore

  • The Unit Circle — All six trig ratios read off one rotating point.

Real-world applications

  • Waves & oscillations: SHM is uniform circular motion's shadow.
  • Exact-value tables: 0°, 30°, 45°, 60°, 90° — read off the circle.
  • Phasors in AC circuits.

JEE exam tips

  • sin(90° ± θ) → cos θ family: co-function flips at odd multiples of 90°, sign from ASTC.
  • The exact-value ladder: sin = √0/2, √1/2, √2/2, √3/2, √4/2 for 0°→90°.
  • tan blows up where cos = 0: vertical asymptotes at 90° + 180°k.

Common mistakes

  • Placing sin on the x-axis (sin is the HEIGHT).
  • Sign errors in Q3/Q4 — draw the point, don't memorise blindly.
  • Degrees vs radians in calculators & calculus.

Exam traps to avoid

  • sin is odd, cos is even: sin(−θ) = −sin θ, cos(−θ) = cos θ.
  • sec/cosec are NEVER in (−1, 1) — range questions exploit this.