Sine and cosine on the unit circle
"Sine and cosine on the unit circle" is a free, interactive Class 11 Maths simulation for JEE, built on the lesson "Sine and cosine on the unit circle" in Trigonometric Functions. Slide x to turn P round the unit circle; negative x turns it clockwise. The blue segment is cos x, the x-coordinate of P, and the green one is sin x, its y-coordinate. The graph on the right follows the value through one turn. P = (cos x, sin x) for every real x, so cos² x + sin² x = 1 and both values stay between −1 and 1. Adding 360° brings P back to the same place: the values repeat every 2π. JEE Main asked 41 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Sine and cosine on the unit circle in Trigonometric Functions. Change the values and watch what happens.
In JEE Main
42 shifts analysed
41 questions from Trigonometric Functions across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §3.3
- Put the unit circle's centre at the origin and start at A(1, 0). Turn through an angle x (in radians) to reach P(a, b) on the circle. Then cos x = a and sin x = b: cosine is the x-coordinate of P and sine the y-coordinate.
- Because P lies on the unit circle, a² + b² = 1, which gives cos² x + sin² x = 1 for every real x.
- For acute x these agree with the triangle ratios of earlier classes, but the unit-circle definition works for every real x, including obtuse, reflex and negative angles.
- The quadrantal angles are the integer multiples of π/2. At x = 0, π/2, π and 3π/2 the point P is (1, 0), (0, 1), (−1, 0) and (0, −1), so cos x takes the values 1, 0, −1, 0 and sin x the values 0, 1, 0, −1.
- A further turn of 2π brings P back to the same point, so sin (2nπ + x) = sin x and cos (2nπ + x) = cos x for every integer n. For example, sin 750° = sin (720° + 30°) = sin 30° = 1/2.
- sin x = 0 exactly when x is an integer multiple of π, that is x = nπ, n ∈ Z.
- cos x = 0 exactly when x is an odd multiple of π/2, that is x = (2n + 1)π/2, n ∈ Z.
Questions
3 answers
- What does the "Sine and cosine on the unit circle" simulation show?
- Slide x to turn P round the unit circle; negative x turns it clockwise. The blue segment is cos x, the x-coordinate of P, and the green one is sin x, its y-coordinate. The graph on the right follows the value through one turn.
- Does JEE Main ask questions from Trigonometric Functions?
- Yes. JEE Main asked 41 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Sine and cosine on the unit circle" in Trigonometric Functions (Class 11 Maths).
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