Playbook
Trigonometric Functions
312 q - 4.46/paper - 41% HARD. The Std XII trigonometry chapter and the second-largest in the bank, in three parts over six notes pages: equations (69 q), solution of triangle (107 q: the rules, then half-angle and area) and inverse trigonometry (136 q: values, identities, equations). The parts cost about the same; the one expensive page is inverse-trig identities at 65% HARD. Solution of triangle is rising (1.26 to 1.85 a paper from 2023-24 to 2025-26) while equations are falling (1.33 to 0.85).
- Questions in the bank
- 312
- q/paper (2025–26 shifts)
- 4.46
- Tagged HARD
- 41%
- Subtopics
- 6
Strand: Cornerstone
When you’ll see it
A triangle labelled with sides a, b, c and angles A, B, C — or an expression built from arcsin, arccos or arctan.
How this chapter is tested
312 q, 4.46/paper, 41% HARD — the second-largest chapter in the bank, behind only Vectors. It is the Std XII trigonometry chapter, and the reason students under-prepare it is that the name is unfamiliar: the chapter they recognise from Std XI, Trigonometry - II, is 61 questions of identities, a fifth of this one.
The chapter eased from 4.93 q/paper across the 27 shifts of 2023-24 to 4.46 across the 39 shifts of 2025-26, third on the paper behind Line and Plane and Vectors, and its three parts are moving differently. Solution of triangle climbed from 1.26 to 1.85 a paper, while inverse trigonometry fell from 2.33 to 1.77 and trigonometric equations from 1.33 to 0.85. The equations sat in a separate Std XI chapter until 2026-09-26, when the bank was re-carved to the Balbharati Std XII syllabus, which puts them here.
The three parts cost almost the same — equations 69 q at 42% HARD, solution of triangle 107 q at 40%, inverse trigonometry 136 q at 41% — so the chapter is drilled whole. The notes split it into six pages, and the difficulty is not even across them: inverse-trig identities (sums, substitution, telescoping) is 65% HARD, while inverse-trig values (24%) and inverse-trig equations (31%) sit well under it. Bank those two before the identities.
Every part is a closed list. The triangle work is four rules and their area forms, and recognition is most of the skill — what you are given (three sides, two sides and the included angle, two angles and a side) decides which rule opens the question. The inverse work is almost entirely about staying inside the principal branch: the algebra is short, and 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.
Trigonometric Equations and General Solutions
The three general-solution patterns (sine takes (-1)^n, cosine takes plus-or-minus, tangent neither), quadratics in one ratio with the impossible or undefined root rejected, a cos x + b sin x = c through its range, factorising sums of sines, and range arguments that show there is no solution.
Solution of Triangle — Sine, Cosine and Projection Rules
a / sin A = 2R, so angles in a ratio give sides in the ratio of their SINES; the cosine rule forwards and backwards, and every 'find the angle' relation among the sides read as a^2 + b^2 - c^2 = k ab; the projection rule for sums of sides times cosines.
Solution of Triangle — Half-Angle Formulas, Napier's Analogy and Area
tan(A/2) tan(C/2) = (s - b)/s and cot(B/2) cot(C/2) = s/(s - a); sides in A.P. make the first product 1/3. Napier's analogy for the difference of two angles, the right-triangle quadratic relation p + q = r, and Heron's formula.
Inverse Trigonometric Functions — Principal Values and Evaluation
The six principal ranges, cold: arcsin in [-pi/2, pi/2], arccos in [0, pi], arctan in (-pi/2, pi/2). cos^-1(-x) = pi - cos^-1 x, not a negative angle. A ratio of an inverse value is read off a right triangle.
Inverse Trigonometric Identities — Sums, Substitution and Telescoping
arcsin x + arccos x = pi/2 and its cousins; arctan x + arctan y = arctan((x + y)/(1 - xy)) while xy < 1; the substitutions x = tan t and x = cos 2t; and the split arctan((A - B)/(1 + AB)) = arctan A - arctan B that makes long sums telescope. The hardest page of the chapter.
Inverse Trigonometric Equations
Combine with a complementary pair or the addition formula, take a trig function of both sides, solve — then substitute every root back. The questions that ask how many elements a solution set has are testing that last check.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
R instead of 2R
a / sin A equals 2R, not R. The distractor is exactly half or exactly double the correct circumradius.
The ambiguous case
Two sides and a non-included angle can determine two different triangles. The option offering a single answer where two are valid — or the one quoting the obtuse solution when only the acute is admissible — is the planted one.
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.
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.
Degrees where radians are meant
Principal values are stated in radians. An option set mixing pi/6 with 30 is signalling this trap.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Trigonometric Functions notesDrill every trigonometric functions question
312 questions from the bank, scoped to 6 bundled subtopics.
Drill one subtopic at a time
The 6 subtopics this playbook covers, in catalog order.
- Trigonometric Equations and General SolutionsDrill
- Solution of Triangle — Sine, Cosine and Projection RulesDrill
- Solution of Triangle — Half-Angle Formulas, Napier's Analogy and AreaDrill
- Inverse Trigonometric Functions — Principal Values and EvaluationDrill
- Inverse Trigonometric Identities — Sums, Substitution and TelescopingDrill
- Inverse Trigonometric EquationsDrill
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