PYQ Vault

Playbook

Inverse Trigonometric Functions

Every question turns on the principal ranges; sums of inverse tangents often telescope.

Questions in the bank
70
q/paper in 2025–26
0.49
Numeric answer
23%
Notes pages
5

Tier: Long tail

When you’ll see it

sin⁻¹, cos⁻¹ or tan⁻¹ of a number or an expression, a sum of inverse tangents, or an equation in inverse functions.

How this chapter is tested

Every question here turns on the principal ranges: sin⁻¹ in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2). The first pages evaluate and simplify inside those ranges — sin⁻¹(sin 3) = π − 3, a ratio read off a right triangle, an expression in x simplified by putting x = sin θ or tan θ.

The later pages put the tools to work: sums of inverse tangents, often a telescoping series whose general term splits as tan⁻¹(next) − tan⁻¹(this), and equations whose roots must be checked against the ranges. Some questions want a numerical answer, where a lost sign has no option to expose it.

Time goes on branch choices. sin⁻¹(2x√(1 − x²)) and tan⁻¹ a + tan⁻¹ b each change formula outside an interval, and squaring an equation brings in false roots. The same simplifications shorten derivatives in Differentiation, and the telescoping idea is the one used in Sequences and Series.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Domain, range and principal values

    Where each inverse is defined, what values it takes, and how sin⁻¹(sin x) is brought back into the principal range.

  • Values of inverse expressions

    Read each angle off a right triangle, then use the double-, half- or triple-angle formulas; take the sign from the range.

  • Simplifying by substitution

    Put x = sin θ, cos θ or tan θ, track the interval of θ, and use sin⁻¹x + cos⁻¹x = π/2.

  • Sums and telescoping series

    tan⁻¹ a + tan⁻¹ b = tan⁻¹((a + b)/(1 − ab)) when ab < 1; write each term of a series as a difference of two inverse tangents.

  • Equations

    Combine terms, take a tangent, sine or cosine, solve, then check every root in the original equation.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • sin⁻¹(sin x) is not always x

    sin⁻¹(sin 3) = π − 3, because 3 lies outside [−π/2, π/2]. Check the answer lies in the principal range.

  • When the product exceeds 1

    For positive a and b with ab > 1, tan⁻¹ a + tan⁻¹ b = π + tan⁻¹((a + b)/(1 − ab)). The bare formula gives a negative angle.

  • Taking tangents or squaring adds roots

    Every root of the resulting polynomial must be put back into the original equation; a negative root often fails.

  • A triangle has no signs

    cos⁻¹ of a negative number lies in (π/2, π), so its cosine and tangent are negative even though the triangle gives positive ratios.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Inverse Trigonometric Functions notes

Drill every Inverse Trigonometric Functions question

70 questions from the bank, across 5 subtopics.

Drill the 70 questions

Drill one subtopic at a time

The 5 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.