Playbook
Trigonometric Functions
207 q - 5.08/paper - 38% HARD. The Std XII trigonometry chapter and the second-largest in the bank, in three parts over six notes pages: equations (47 q), solution of triangle (69 q: the rules, then half-angle and area) and inverse trigonometry (91 q: values, identities, equations). The parts cost about the same; the one expensive page is inverse-trig identities at 63% HARD. Solution of triangle is rising (1.64 to 2.13 a paper) while equations are falling (1.12 to 0.83).
- Questions in the bank
- 207
- q/paper (2024–25 shifts)
- 5.08
- Tagged HARD
- 38%
- 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
207 q, 5.08/paper, 38% 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 39 questions of identities, a fifth of this one.
The chapter edged up from 4.93 q/paper across the lifetime window to 5.08 across the 24 shifts of 2024-2025, the heaviest rate on the paper, and its three pages are moving differently. Solution of Triangle climbed from 1.64 to 2.13, inverse trigonometry held (2.17 to 2.13), and trigonometric equations fell from 1.12 to 0.83. 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 47 q at 36% HARD, solution of triangle 69 q at 39%, inverse trigonometry 91 q at 37% — 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 63% HARD, while inverse-trig values and inverse-trig equations are both under 25%. 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
207 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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