Does sin⁻¹ undo sin?
"Does sin⁻¹ undo sin?" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Undoing an inverse" in Inverse Trigonometric Functions. The dashed line is y = θ. The solid zigzag is y = sin⁻¹(sin θ). Inside the shaded band they agree. Outside it, the inverse can only answer with an angle in the band, so the graph folds back. F(f⁻¹(x)) = x whenever f⁻¹(x) exists, but f⁻¹(f(θ)) = θ only for θ in the principal range. Outside it, swap θ for the angle in the range with the same value of f. JEE Main asked 25 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Undoing an inverse in Inverse Trigonometric Functions. Change the values and watch what happens.
In JEE Main
42 shifts analysed
25 questions from Inverse Trigonometric Functions across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §2.3
- From the definition: sin(sin⁻¹ x) = x for every x ∈ [−1, 1], and sin⁻¹(sin x) = x for x ∈ [−π/2, π/2]. The same pairs hold for the other five functions on their own domains and principal ranges.
- The first form is always safe inside the domain: cos(cos⁻¹ 0.3) = 0.3 and tan(tan⁻¹ 5) = 5.
- The second form fails outside the principal range, because the inverse can only return a principal angle. sin⁻¹(sin 2π/3) is π/3, not 2π/3.
- Method: replace the angle by one in the principal range with the same trigonometric value. sin 3π/5 = sin(π − 3π/5) = sin 2π/5, and 2π/5 lies in [−π/2, π/2], so sin⁻¹(sin 3π/5) = 2π/5.
- tan⁻¹(tan 3π/4): tan 3π/4 = −1, and tan⁻¹(−1) = −π/4. cos⁻¹(cos 7π/6): cos 7π/6 = −√3/2 and cos⁻¹(−√3/2) = 5π/6.
- Drawn as a function of x over all of R, y = sin⁻¹(sin x) is a zigzag: it follows y = x on [−π/2, π/2], then y = π − x on [π/2, 3π/2], and so on, never leaving [−π/2, π/2].
- Mixed forms reduce to a right triangle. If θ = tan⁻¹ x with |x| < 1, the triangle has opposite x, adjacent 1 and hypotenuse √(1 + x²), so sin(tan⁻¹ x) = x/√(1 + x²).
Questions
3 answers
- What does the "Does sin⁻¹ undo sin?" simulation show?
- The dashed line is y = θ. The solid zigzag is y = sin⁻¹(sin θ). Inside the shaded band they agree. Outside it, the inverse can only answer with an angle in the band, so the graph folds back.
- Does JEE Main ask questions from Inverse Trigonometric Functions?
- Yes. JEE Main asked 25 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 "Undoing an inverse" in Inverse Trigonometric Functions (Class 12 Maths).
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