Playbook
Inverse Trigonometric Functions
73 q - 2.04/paper - 37% HARD. A single-subtopic chapter, so there is nothing to cherry-pick: it is one 73-question block of identities, equations, principal values and sums. Its weightage has RISEN (1.66 lifetime to 2.04 recent).
- Questions in the bank
- 73
- q/paper (2024–25 shifts)
- 2.04
- Tagged HARD
- 37%
- Subtopic
- 1
Strand: Long Tail
When you’ll see it
An expression built from arcsin, arccos or arctan — a sum to be collapsed, an equation to be solved, or a principal value to be named.
How this chapter is tested
73 q in a single subtopic, 2.04/paper, 37% HARD. Being one undivided block means there is nothing to cherry-pick, but it also means the chapter has one syllabus rather than three: identities, equations, principal values and sums, all drawn from the same short closed list. At 37% HARD it is materially cheaper than Limits at 56% or Trigonometry - II at 49%.
It is one of the few chapters moving in the right direction. Its weightage has RISEN from 1.66 lifetime to 2.04 recent — the opposite of Trigonometry - I, which has fallen from 2.20 to 1.85. A student prioritising from lifetime frequency alone under-invests here and over-invests there.
Add the overlap and the chapter is bigger than its headline. Trigonometry - II carries a further 21 inverse-trig questions at 52% HARD, so the real target is 94 questions. This chapter is the softer 73 of them; treat the Trigonometry - II subtopic as the hard tail of the same topic rather than as separate material.
The work is almost entirely about staying inside the principal branch. The algebra is short; the marks are lost by producing a technically valid value that lies outside the allowed range.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Principal-value ranges, cold
arcsin lands in [-pi/2, pi/2], arccos in [0, pi], arctan in (-pi/2, pi/2). Nothing else in this chapter is safe until these three are automatic.
Complementary identities
arcsin x + arccos x = pi/2, arctan x + arccot x = pi/2, arcsec x + arccosec x = pi/2. These collapse a whole family of sum questions to a single constant.
Sum and difference of arctan
arctan x + arctan y = arctan((x + y)/(1 - xy)), valid only while xy is less than 1. Outside that, add or subtract pi to land back in the principal branch — the side condition is the point of the formula, not a footnote.
Converting between inverse functions
Rewriting arcsin as arctan, or arccos as arcsin, using a right triangle drawn from the given ratio. This is what makes a mixed-function expression collapse.
Solving inverse-trig equations
Take the appropriate trig function of both sides, solve the resulting algebraic equation, then discard every root whose corresponding value falls outside the principal range. The discard step is the marked one.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
The arctan sum without its condition
The unadjusted arctan((x + y)/(1 - xy)) is always offered. When xy exceeds 1 the true answer differs from it by pi, and the raw value is the distractor.
Range violation
An algebraically correct value outside the principal branch. Both it and the corrected value appear in the option set; only one lies in range.
arcsin(sin x) assumed to be x
True only for x in [-pi/2, pi/2]. Outside that range the answer is pi - x or x minus a multiple of 2 pi, and the option reading plain x is the trap.
Domain of the argument ignored
arcsin and arccos accept only inputs in [-1, 1]. A question engineered so that a careless substitution produces an argument outside that range is testing whether you checked.
Drill every inverse trigonometric functions question
73 questions from the bank, scoped to the named subtopic.
Drill one subtopic at a time
The 1 subtopic this playbook covers, in catalog order.
- Inverse Trigonometric Functions — Identities, Equations, Principal Values, and SumsDrill
Related playbooks
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