Playbook
Trigonometry - II
90 q - 2.08/paper - 49% HARD. Dominated by Properties of Triangles (52 q, sine, cosine and projection rules). Note the taxonomy overlap: this chapter carries an Inverse Trigonometry subtopic (21 q) that duplicates the standalone Inverse Trigonometric Functions chapter - drill both or you will miss half the inverse-trig questions.
- Questions in the bank
- 90
- q/paper (2024–25 shifts)
- 2.08
- Tagged HARD
- 49%
- Subtopics
- 3
Strand: Long Tail
When you’ll see it
A triangle labelled with sides a, b, c and angles A, B, C — or an inverse-trig expression turning up inside a chapter you were not expecting it in.
How this chapter is tested
90 q, 2.08/paper, 49% HARD. The chapter is dominated by Properties of Triangles — Sine/Cosine Rules and Projection, 52 of the 90 questions at 42% HARD. That block is both the largest and the least punishing part of the chapter, and it is where preparation should go first.
The other two subtopics are small and expensive: Inverse Trigonometry sits at 52% HARD across 21 q, and Trigonometric Identities and Compound/Half-Angle Formulas at 65% HARD across 17 q — the second-highest HARD rate of any subtopic in the long tail. With the whole chapter running about two questions a paper, that 17-q corner is the honest place to give a question your best guess and move on.
The measured taxonomy overlap matters here more than anywhere else on the paper. Inverse trigonometry is filed BOTH as the 21-q subtopic in this chapter and as its own 73-q chapter, so the bank holds 94 inverse-trig questions in two places. Drill only the standalone chapter and you skip 21 of them — and they are the harder 21, at 52% against the standalone chapter's 37%.
The triangle work is closed and reusable: sine rule, cosine rule, projection formula, and the area forms. Recognition is most of the skill — decide from what you are given (three sides, two sides and the included angle, two angles and a side) which rule opens the question.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Sine rule and circumradius
a / sin A = b / sin B = c / sin C = 2R. The 2R is the half most often dropped, and it is what turns a triangle question into a circumradius question.
Cosine rule, both directions
Use it forwards to find a side from two sides and the included angle, and backwards, as cos A = (b^2 + c^2 - a^2) / 2bc, to find an angle from three sides.
Projection formula
a = b cos C + c cos B and its two cousins. Cheap to memorise and it collapses a family of 'prove or simplify' questions in one line.
Area and half-angle forms
Area = (1/2) ab sin C, Heron's form with the semi-perimeter s, and the relations linking area to the inradius r and circumradius R. These are the bridge between a triangle question and a circle question.
Inverse-trig block
Principal-value ranges, the complementary identities, and the sum formulas with their side conditions. Shared verbatim with the Inverse Trigonometric Functions playbook — learn it once, it answers 94 questions across the two chapters.
Compound and half-angle identities
sin(A plus or minus B), cos(A plus or minus B), tan(A plus or minus B), and the half-angle and multiple-angle forms. This is the chapter's 65% HARD corner and the last thing to reach.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
The overlap itself
Treating this chapter as triangles-only. A fifth of its questions are inverse trigonometry, and a study plan built on the chapter name alone walks past them.
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.
Degrees where radians are meant
Principal values in the inverse-trig subtopic are stated in radians. An option set mixing pi/6 with 30 is signalling this trap.
Drill every trigonometry - ii question
90 questions from the bank, scoped to 3 bundled subtopics.
Drill one subtopic at a time
The 3 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.