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Math notes / Trigonometry

Trigonometry · 100% marks notes

Trigonometry becomes exam-ready when you open the right topic page: Formulas for reduce, Equations for general solutions, Inverse for principal values, Triangles for sine/cosine-rule joins — each with its own 60–90s process.

Goal Bank formulas + equations + inverse; keep SOT ready for joinsWeight Official Trig unit + frequent Calculus/Complex joins; SOT as join toolkitSpeed 60–90sShare 10.3%Trend Stable
All Math notesAlgebraCalculusCoordinateTrigVectors · 3D
Jump to topicFormulasEquationsInverseTriangles
Trig is no longer one bundled page. Study four independent topics: (1) Formulas & applications, (2) Trigonometric equations, (3) Inverse trigonometry, (4) Solution of triangles (join tool — not a guaranteed standalone Advanced chapter). This hub opens with the official Trig syllabus and 2020–2026 trends, then the shared 60–90s protocol. Trig full marks come from job naming — formula, equation, inverse, or triangle.
Topic-wise on this page

Trigonometry · chapter details

Same content as each topic page — brief, paper asks, methods, formulas, and families — so you can study here first, then open a full topic only when you need samples or trends.

01 · Foundation for every Trig stem + Calculus/Complex joins

Trig Formulas & Applications

Full topic page →

PromiseSin / cos / tan marks come from a fixed toolkit: A±B → multiple/sub-multiple → factor (sum↔product) → apply to a value or simplify — never from re-deriving identities mid-paper.

This page is the formula engine: periodicity & graphs, addition/subtraction, multiple and sub-multiple angles, factor formulae, and how Advanced actually uses them (exact values, limits, integrals, complex polar). Pair with Equations and Inverse for full Trig. Trig formulae Advanced is job-naming plus bankable exact values — not memorising every line.

01. Why this page exists alone: bundling ‘all Trig’ made students skip compound-angle fluency. Advanced rarely asks ‘prove the identity’ — it asks you to reduce an expression in 20s so Calculus or an equation can finish.

02. Official anchors: trig functions, periodicity and graphs, addition/subtraction, multiple and sub-multiple formulae. Graphs tell you range and zeros; formulae turn products into sums and vice versa.

03. Application mindset: every formula has a job — expand (A±B), compress (factor), double/half (multiple), evaluate (standard angles). Name the job before picking the line.

How Advanced asks this

  • Simplify sin/cos/tan expressions to a single term or constant
  • Exact value of sin(π/12), cos(15°), tan(A±B) chains
  • Convert product ↔ sum before integrating or taking limits
  • MC identity T/F with a tiny angle test
  • Joins: lim sinx/x, ∫sin², complex Arg / De Moivre

Traps that leak marks

  • Wrong sign in cos(A−B) vs cos(A+B)
  • Ignoring tan undefined (1−tanA tanB=0)
  • Expanding everything when factor would compress
  • Forgetting power-reduce in ∫sin²
  • Expanding a sum that factors with a product-to-sum in one line
  • Wrong sign in cos(A−B) or tan(A+B) denominator
  • Accepting an identity that fails at a tiny angle plug

Must cover (this topic)

  • Multiple/sub-multiple angle toolkit

60–90s process

  1. 0–15s — Name ask: simplify / exact value / product–sum / graph-range. Circle angles.
  2. 15–35s — Pick ONE identity family and rewrite. Prefer factor for sums; prefer double for 2A forms.
  3. 35–75s — Finish algebra; for applications, feed the reduced form into limit/integral/equation.
  4. 75–90s — Sanity: value in [−1,1] for sin/cos; check a special angle; reject if tan denominator is 0.

Methods that change the score

Formulas are a job list: expand, compress, double, evaluate. These methods stop random identity shopping and make π/12 and factor moves automatic.

Method 01 · Name the identity job

Stops random formula shopping — expand/compress/double/evaluate becomes automatic.

Do this: Say expand / compress / double / evaluate before writing algebra.

Method 02 · π/12 via 45±30

Exact 15°/75° values become bankable instead of guesswork.

Do this: For 15°/75°, always use 45°±30° compound — never invent new angles.

Method 03 · Factor before expand-all

Sums that look impossible compress in one line.

Do this: Sums of sin/cos → factor formulae first; only expand if factor fails.

Method 04 · Tiny-angle identity test

False MC identities die in seconds — toppers use this constantly.

Do this: On opaque MC identities, plug x=π/6 or π/4 to kill false options.

Quick formulas

  • sin(A±B) / cos(A±B): sin(A±B)=sinA cosB±cosA sinB; cos(A±B)=cosA cosB∓sinA sinB — Compound expand/reduce
  • tan(A±B): tan(A±B)=(tanA±tanB)/(1∓tanA tanB) — Tangent compound; check 1∓tanA tanB≠0
  • Double angle: sin2A=2sinA cosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A) — Multiple-angle compress/expand
  • Half / sub-multiple: sin(A/2)=±√((1−cosA)/2); cos(A/2)=±√((1+cosA)/2); tan(A/2)=sinA/(1+cosA) — Half-angle; sign from quadrant
  • Factor (sum→product): sinC+sinD=2sin((C+D)/2)cos((C−D)/2); cosC+cosD=2cos((C+D)/2)cos((C−D)/2) — Sum of sines/cosines
  • Product→sum: 2sinA cosB=sin(A+B)+sin(A−B); 2cosA cosB=cos(A+B)+cos(A−B) — Products before integrate/limit
  • sin² / cos² power-reduce: sin²A=(1−cos2A)/2; cos²A=(1+cos2A)/2 — Integrals and averages
  • Periodicity (core): sin(x+2π)=sin x; cos(x+2π)=cos x; tan(x+π)=tan x — Graphs, range, general solutions setup

Common question families

Exact value via A±B

Looks like: sin/cos/tan of 15°, 75°, or compound angles

Fast model: Split into 45±30 (or other known) → expand A±B → simplify.

Factor / product–sum

Looks like: Simplify sinC±sinD; convert products before integrate/limit

Fast model: Name compress vs product→sum; apply one identity; stop.

Power-reduce / double angle

Looks like: sin², cos², sin2A forms in simplify or Calculus joins

Fast model: sin²A=(1−cos2A)/2; spot 2A forms early.

Conditional identity / prove for all x

Looks like: Prove identity; find condition so identity holds; MATCH left↔right

Fast model: Move all to one side → factor / express in one function → verify ≡0.

Sample preview · Exact value via A±B

Find the exact value of sin 75°.

Answer: (√6+√2)/4

π/12 family always via 45±30.

More samples on full topic page →

Open full Formulas notes (samples · trends · joins) →

02 · Official Advanced core — general solutions every few papers

Trigonometric Equations

Full topic page →

PromiseEquation full marks = reduce to one function → find α → stamp the correct general template (sin/cos/tan) → reject extraneous roots. Listing only [0,2π] when ℝ is asked is a mark killer.

Official Advanced: general solution of trigonometric equations, with periodicity/graphs as the foundation. This page is only equations — use Formulas for the identity toolkit that feeds the reduce step. Trig equations under pressure are reduce → template → extraneous check.

01. Why equations decide Trig marks: SC/MATCH often ask the full solution set on ℝ. α correct but period wrong still loses the question.

02. Graph view: sin and cos repeat every 2π; tan every π. The general solution is the graph’s full family of intersections, not one period’s roots.

03. Process: identity reduce → principal α → template → plug-check if you squared or used tan.

How Advanced asks this

  • Solve sin/cos/tan equation for x∈ℝ
  • Find number of solutions in [0,2π] or another interval
  • MATCH equation ↔ solution set
  • Equations after squaring or t=tan(x/2)
  • Joins: range of solutions feeding into Calculus conditions

Traps that leak marks

  • Only listing [0,2π] when ℝ asked
  • Wrong period (π vs 2π)
  • Forgetting ± in cos general solution
  • Extraneous roots after squaring
  • Losing solutions when dividing by sin/cos without checking zeros
  • Extraneous roots after squaring
  • Using sin-general-solution on a cos-reduced equation

Must cover (this topic)

  • General solution of trig equations

60–90s process

  1. 0–15s — Name: general ℝ / count in interval / MATCH. Circle target function.
  2. 15–35s — Reduce with Formulas toolkit to sin θ = k, cos θ = k, or tan θ = k (or a quadratic in one of them).
  3. 35–75s — Find α; apply sin/cos/tan general template; if interval asked, list n that fit.
  4. 75–90s — Plug back after squaring; check n=0,1 sanity; for MC tick each option vs template.

Methods that change the score

Equation marks are template marks. These methods force reduce → α → general form — and kill squaring ghosts before they become options.

Method 01 · Reduce to one function

Turns multi-trig mess into a template case you already trust.

Do this: Expand/factor until one of sin/cos/tan (or a quadratic in one) remains.

Method 02 · General solution template

α correct but period wrong still loses marks — this locks the scored form.

Do this: Find α, stamp the matching nπ / 2nπ form, write n∈ℤ.

Method 03 · R-method for a sin + b cos

One harmonic form instead of expanding forever.

Do this: Rewrite as R sin(θ+α) before solving.

Method 04 · Check extraneous roots

Squaring ghosts are designed into options; plug-back is mandatory.

Do this: After squaring or Weierstrass, plug every candidate back.

Quick formulas

  • General (sin): sin θ = sin α ⇒ θ = nπ + (−1)ⁿ α, n∈ℤ — sin equations
  • General (cos): cos θ = cos α ⇒ θ = 2nπ ± α, n∈ℤ — cos equations
  • General (tan): tan θ = tan α ⇒ θ = nπ + α, n∈ℤ — tan equations (where defined)
  • sin θ = k: If |k|≤1, α=sin⁻¹k (principal), then sin template — Numeric RHS
  • cos θ = k: If |k|≤1, α=cos⁻¹k, then cos template — Numeric RHS
  • Linear a sinθ + b cosθ: R sin(θ+α) or R cos(θ−α) with R=√(a²+b²) — Single harmonic form before solving
  • No solution gate: |sin| or |cos| > 1 ⇒ empty set — Quick reject

Common question families

General solution of equations

Looks like: Solve sin/cos/tan = k; find all real x; MATCH equation ↔ set

Fast model: Find α → stamp matching general template → check n=0,1.

a sin + b cos = c

Looks like: Linear combination equal to a constant

Fast model: R-method → R sin(θ+α)=c → general solution.

Extraneous after squaring

Looks like: Equation after squaring; Weierstrass high degree

Fast model: Solve transform → plug every candidate back.

Count solutions in an interval

Looks like: Number of solutions in [0,2π) / (0,π]; how many roots

Fast model: Write full general solution → restrict n so x lands in the interval → count distinct.

Sample preview · General solution

Solve sin x = 1/2 for x ∈ ℝ.

Answer: x = nπ + (−1)ⁿ(π/6), n∈ℤ

Always write the full template after α.

More samples on full topic page →

Open full Equations notes (samples · trends · joins) →

03 · Official Advanced — principal values only; high MC trap density

Inverse Trigonometry

Full topic page →

PromiseInverse Trig full marks = lock principal range first, then evaluate or apply elementary identities. Same discipline as Arg in Complex — wrong branch is a designed distractor.

Official Advanced: inverse trigonometric functions (principal value only) and elementary properties. This page is only Inverse — ranges, standard values, sum/difference identities, and composition with trig. Inverse trig marks follow range locks — formulae without range are traps.

01. Why Inverse is separated: students confuse multi-valued inverse with principal value. Advanced marks the principal branch only.

02. Ranges to automate: sin⁻¹∈[−π/2,π/2], cos⁻¹∈[0,π], tan⁻¹∈(−π/2,π/2), cot⁻¹ usually (0,π), sec⁻¹/csc⁻¹ per your textbook convention — write yours on rough sheet before options.

03. Elementary properties: sin⁻¹x+cos⁻¹x=π/2, tan⁻¹x+cot⁻¹x=π/2 (with domain care), and composition identities like sin(sin⁻¹x)=x on [−1,1].

How Advanced asks this

  • Principal value of sin⁻¹/cos⁻¹/tan⁻¹ of a number
  • Simplify sin⁻¹x+cos⁻¹x or tan⁻¹a±tan⁻¹b
  • MC: which value is principal?
  • Composition: sin(cos⁻¹x), tan(sin⁻¹x)
  • Joins: inverse inside integrals or as substitution

Traps that leak marks

  • Non-principal branch (negative cos⁻¹)
  • Using π/2 identity outside |x|≤1
  • Arctan sum without ab>1 adjustment
  • Wrong sign on √(1−x²) compositions
  • tan⁻¹a+tan⁻¹b formula when ab>1 without π correction
  • Dropping absolute value / sign on √(1−x²) compositions
  • Using an identity outside the principal range

Must cover (this topic)

  • Inverse trig principal values

60–90s process

  1. 0–15s — Underline which inverse; write its principal range immediately.
  2. 15–35s — Evaluate reference angle; adjust sign/quadrant to land inside principal range.
  3. 35–75s — Apply identity only after values are principal-safe; watch tan⁻¹a+tan⁻¹b when ab>1.
  4. 75–90s — Verify answer ∈ principal range; reject lookalike 2π-shifts.

Methods that change the score

Inverse Advanced is principal-range discipline. These methods lock range first — same habit as Arg — so wrong-branch distractors die.

Method 01 · Principal inverse lock

Same habit as Arg in complex: range first, value second — distractors die.

Do this: Write the range of the inverse function before picking a value.

Method 02 · Triangle for composition

sin(cos⁻¹x) style asks stop being sign roulette.

Do this: For sin(cos⁻¹x), draw right triangle with adjacent=x, hypotenuse=1.

Method 03 · Identity only after range-safe

π/2 identities without |x|≤1 create confident wrong answers.

Do this: Do not apply sin⁻¹x+cos⁻¹x=π/2 until |x|≤1 is clear.

Method 04 · Arctan ab>1 gate

The classic tan⁻¹a+tan⁻¹b trap dies before you write the wrong formula.

Do this: When adding tan⁻¹a+tan⁻¹b, check ab vs 1 before the simple formula.

Quick formulas

  • Principal ranges: sin⁻¹∈[−π/2,π/2]; cos⁻¹∈[0,π]; tan⁻¹∈(−π/2,π/2) — Any inverse value
  • sin⁻¹x + cos⁻¹x: sin⁻¹x + cos⁻¹x = π/2 for x∈[−1,1] — Sum identity
  • tan⁻¹x + cot⁻¹x: tan⁻¹x + cot⁻¹x = π/2 (principal conventions) — Complementary pair
  • tan⁻¹a ± tan⁻¹b: tan⁻¹a+tan⁻¹b=tan⁻¹((a+b)/(1−ab)) when ab<1 (adjust if ab>1) — Sum/difference of arctan
  • sin(sin⁻¹x): sin(sin⁻¹x)=x on [−1,1]; cos(cos⁻¹x)=x on [−1,1] — Direct composition
  • sin(cos⁻¹x): sin(cos⁻¹x)=√(1−x²) (non-negative for principal cos⁻¹) — Triangle / Pythagorean composition
  • Standard values: sin⁻¹(1/2)=π/6; cos⁻¹(1/2)=π/3; tan⁻¹1=π/4; cos⁻¹(−1/2)=2π/3 — Fast MC

Common question families

Principal values

Looks like: sin⁻¹/cos⁻¹/tan⁻¹ of a number; which option is principal?

Fast model: Lock range → evaluate reference → land inside range.

Sum / complementary identities

Looks like: sin⁻¹x+cos⁻¹x; tan⁻¹a±tan⁻¹b

Fast model: Check domain; for arctan check ab vs 1; then identity.

Composition via triangle

Looks like: sin(cos⁻¹x), tan(sin⁻¹x)

Fast model: Draw right triangle from the inner inverse; read the ratio; watch sign.

Equation in inverse trig → principal solve

Looks like: Solve sin⁻¹x + cos⁻¹y = …; find x from an inverse equation

Fast model: Range lock → apply identity or take sin/cos of both sides carefully → reject extras.

Sample preview · Principal value

Find the principal value of sin⁻¹(−1/2) + cos⁻¹(1/2).

Answer: π/6

Never pick the non-principal branch.

More samples on full topic page →

Open full Inverse notes (samples · trends · joins) →

04 · Not a direct Advanced unit — still a join tool inside Vectors/Geometry/Integrals

Solution of Triangles

Full topic page →

PromiseSOT full marks (when it appears as a tool) come from Law of Sines / Cosines + area formulas first — then half-angle / projection only if needed. Treat this as a bridge chapter, not a formula museum.

Direct ‘Solution of Triangles’ was removed from the official Advanced syllabus list, but sine/cosine rules, area, and R=a/(2sinA) still unlock Vector, Coordinate, and integral stems. Study this page as applied toolkit — not as a promise of standalone Advanced questions. Triangles under the clock are case-name first — then Sines, Cosines, or area.

01. Honest syllabus note: Advanced’s printed Trig unit focuses on functions, formulae, equations, and inverse — not classical SOT chapter lists. Still keep SOT sharp for joins.

02. Core toolkit: a/sinA=b/sinB=c/sinC=2R; cosA=(b²+c²−a²)/(2bc); area = (1/2)bc sinA = √[s(s−a)(s−b)(s−c)] = abc/(4R).

03. When it shows up: angle in a triangle of vectors, area via cross product matching (1/2)ab sinC, circumradius in geometry joins.

How Advanced asks this

  • Find a side/angle given ASA/SAS/SSS data (join context)
  • Area of triangle from two sides and included angle
  • Relate R, r, Δ in a geometry/vector stem
  • Ambiguous case SSA awareness (rare but deadly)
  • Hidden SOT inside Vectors or definite integrals

Traps that leak marks

  • Heron on SAS (use (1/2)ab sinC)
  • Ignoring ambiguous SSA
  • Mixing opposite side/angle pairs
  • Treating SOT as guaranteed standalone Advanced marks
  • Heron on SAS when (1/2)ab sinC is immediate
  • One-triangle answer on ambiguous SSA
  • Sines without checking angle opposite the known side pairing

Must cover (this topic)

  • Triangle centres: centroid, orthocentre, incentre, circumcentre

60–90s process

  1. 0–15s — Sketch triangle; mark known sides/angles; name case (SAS/ASA/SSS/SSA).
  2. 15–35s — Choose tool: Sines (ASA/AAS), Cosines (SAS/SSS), area formula if Δ asked.
  3. 35–75s — Compute; keep exact sin/cos of standard angles; introduce 2R if circumradius appears.
  4. 75–90s — Triangle inequality / angle sum 180° check; area > 0.

Methods that change the score

SOT is a join toolkit now. These methods force sketch + case name (SAS/ASA/SSS/SSA) so the right law is chosen in 10 seconds.

Method 01 · Sketch + case name

SAS vs ASA chooses Cosines vs Sines in 10s — that is the whole speed story.

Do this: Draw and label SAS/ASA/SSS/SSA before any formula.

Method 02 · Sines for opposite pairs

Opposite-pair bridge is the fastest path when angle+opposite side exist.

Do this: When an angle and its opposite side are known, use a/sinA bridge.

Method 03 · Area three ways

Stops Heron on SAS data — (1/2)ab sinC is the right tool.

Do this: Prefer (1/2)bc sinA; use Heron only for SSS; use abc/(4R) when R given.

Method 04 · Ambiguous SSA gate

Prevents shipping one triangle when two (or none) exist.

Do this: If SSA, check height h=b sinA before claiming one/two/no triangle.

Quick formulas

  • Law of Sines: a/sinA = b/sinB = c/sinC = 2R — ASA, AAS, or relate side to opposite angle / R
  • Law of Cosines: c² = a²+b²−2ab cosC; cosC=(a²+b²−c²)/(2ab) — SAS, SSS
  • Area (sides + sin): Δ = (1/2)bc sinA — Two sides + included angle
  • Heron: Δ = √[s(s−a)(s−b)(s−c)], s=(a+b+c)/2 — Three sides
  • Δ via R: Δ = abc/(4R) — Circumradius joins
  • Projection: a = b cosC + c cosB — Projection formula applications
  • Half-angle (sin): sin(A/2)=√[((s−b)(s−c))/(bc)] — Half-angle asks (use sparingly)

Common question families

SAS / ASA toolkit

Looks like: Two sides + included angle; or two angles + side

Fast model: Name case → Cosines (SAS) or Sines (ASA) → finish ask.

Area three ways

Looks like: Find Δ from sides/angles; or match vector area

Fast model: (1/2)bc sinA if included angle; Heron if SSS; abc/(4R) if R given.

Sines ↔ 2R join

Looks like: Circumradius; side from opposite angle; vector/geometry join

Fast model: a/sinA=2R bridge.

SSA ambiguous case

Looks like: Two sides + non-included angle; how many triangles

Fast model: Compute height a sin A vs opposite side; decide 0/1/2 triangles before solving.

Sample preview · SAS / ASA toolkit

In △ABC, a=5, b=6, ∠C=30°. Find the area.

Answer: 15/2

Match area formula to the given data — Heron waits for SSS.

More samples on full topic page →

Open full Triangles notes (samples · trends · joins) →

SyllabusTrends · 10.3% · StableMixed joins
Official · 2026

Trigonometry syllabus map

Full Math syllabus →
Maps to Trigonometry unit including inverse trig principal values.

Must cover from syllabus

  • General solution of trig equations
  • Multiple/sub-multiple angle toolkit
  • Inverse trig principal values

Gaps / exclusions to watch

  • Periodicity/graph reading questions

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Trigonometry

medium

Exam focusGeneral solutions; inverse trig principal values; joins with complex and calculus.

Syllabus points

  • Trig functions: periodicity and graphs
  • Addition/subtraction; multiple and sub-multiple angle formulae
  • General solution of trigonometric equations
  • Inverse trigonometric functions (principal value) and elementary properties

Do / check

  • Write principal range of inverse trig before solving
  • Convert to sin/cos single form for equations
Cross-topic

Mixed joins tagged to Trig

All mixed joins →
Mathematics2026 · Paper 2 · Q15–Q16linked-numerical

Exponential curves + area

Calculus – Curves+Definite Integration – Area+Trigonometric equations

Knowledge bridge

Step 1: set y = e^{-x} equal to y = e^{-x}(sin x + cos x) and cancel the common exponential to get a trig equation for intersection abscissae α_i. Step 2: order those roots and set up the definite integral of the difference of the two curves between α_1 and α_4 for the enclosed area β. Step 3: simplify the given log expression in β to a clean numerical value.

Why it feels hard

The stem hides trig solving inside an exponential envelope; students often integrate before finding exact limits.

What to practise

Drill ‘cancel common positive factor → trig roots → area integral’ as one reflex chain.

2026 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2026 · Paper 2 · Q8multi-correct

DE solution + extrema + intersections

Differential Equations+Functions / AOD+Curve intersections

Knowledge bridge

Step 1: solve x y' = y − x³ (homogeneous/linear in y/x) to get the family y = Cx − x³/2. Step 2: impose the given condition to fix C, then use y' = 0 to locate the local max. Step 3: compare monotonicity on (1,2) and count positive roots of f = g.

Why it feels hard

Multi-correct options test DE, calculus and graph reading together; one sign error in the DE kills several options.

What to practise

After every DE, immediately ask: extrema? monotonicity? intersections with a given g?

2026 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2023 · Paper 1 · Q7–Q9multi-concept

Parabola/ellipse tangents + area numerical

Conics – common tangents+Calculus – Area+Trigonometric equations

Knowledge bridge

Step 1: for the ellipse–parabola common-tangent MC, write both tangent conditions and compare. Step 2: later area NUM — find intersection limits first. Step 3: integrate difference of functions; keep trig-equation NUM technique nearby for neighbouring items.

Why it feels hard

Classic Advanced join: geometry conditions feeding calculus numerics in the same paper.

What to practise

Any week you revise conics, end with one area-between-curves NUM.

2023 year review →Mathematics notes →Quick card →Syllabus →
Mathematics2022 · Paper 1 · Q1–Q2numerical

Inverse trig NUM + trig equation NUM

Inverse trigonometry+Trigonometric equations+Numerical answers

Knowledge bridge

Step 1: reduce inverse-trig expressions with principal-value care. Step 2: solve the neighbouring trig equation in the stated interval. Step 3: enter integers carefully — range mistakes are common.

Why it feels hard

Heavy NUM opening stresses accuracy before you are warm.

What to practise

Start Math mocks with a 10-minute trig warm-up when the paper is NUM-heavy.

2022 year review →Mathematics notes →Quick card →Syllabus →
Read first

How to use these notes for 100%

Point 01

Why split Trig: bundling made students skim inverse ranges and skip compound-angle fluency. Advanced rewards specialised habits — formula job naming, general-solution templates, principal locks, and (when needed) triangle case naming.

Point 02

Official Advanced Trig: periodicity/graphs, addition/subtraction, multiple/sub-multiple, general solutions, inverse (principal values). Solution of Triangles is not listed as a direct unit now — keep it for Vectors/Geometry/integral joins.

Point 03

How to read this hub: lock the 60–90s protocol, then open each topic page for methods, samples, and families. Use worked patterns as exam-voice practice.

Point 04

Full marks mindset: pick the topic page that matches the stem in 10s. Wrong page (e.g. Heron on an inverse ask) wastes the budget.

Point 05

Depth upgrade: separate formula-jobs (expand/compress/evaluate) from equation-jobs (general solution) and inverse-jobs (principal range). Mixing those three languages loses marks.

Point 06

SOT is case-named geometry (SAS/ASA/SSA), not a formula dump — sketch first.

How the paper asks

  • Exact values and identity simplifies (Formulas page).
  • General solutions on ℝ / counts in an interval (Equations page).
  • Inverse principal values and compositions (Inverse page).
  • Hidden triangle laws inside Vectors/area joins (Triangles page).
  • Mixed: Trig reduce feeding a limit, integral, or complex Arg.

Scoring plan in the hall

  • Bank clean formula exact-values and standard general solutions early.
  • Never ship inverse without a written principal range.
  • If the stem is clearly SOT-in-disguise, sketch the triangle — do not force pure identity algebra.
  • Protect Calculus joins by keeping A±B and power-reduce automatic.
Speed system

60–90 second answer protocol

Use this on every stem — SC, MC, NUM, MATCH.

Time box

0–15s

Do thisRoute the stem: Formulas / Equations / Inverse / Triangles. Circle the ask.

Why this window matters

Inverse → write range now. Equation → note ℝ vs interval. Triangle → sketch. Formula → name expand/compress/double.

Time box

15–35s

Do thisApply that page’s first method (identity job / reduce / range lock / case name).

Why this window matters

Do not mix pages. Formula reduce feeds Equations — finish reduce before template.

Time box

35–75s

Do thisExecute: expand/factor, general template, principal value, or Law of Sines/Cosines.

Why this window matters

Write n∈ℤ on equation answers. Check ab vs 1 on arctan sums. Use (1/2)ab sinC on SAS area.

Time box

75–90s

Do thisVerify: value ranges, extraneous roots, triangle inequality, principal interval.

Why this window matters

MC: tick each option as independent T/F against your template/range.

Transferable knowledge

Universal methods (all problem types)

Automate these — they apply across years and formats for this portion.

Method 01

Route to the right Trig page

Applies toAll Trig stems

The move

In 10s choose Formulas / Equations / Inverse / Triangles — then open that habit set only.

Why it saves time

Stops using inverse tricks on equation stems and vice versa.

Worked idea

‘Solve sin x = … on ℝ’ → Equations. ‘sin⁻¹(−1/2)’ → Inverse. ‘sin75°’ → Formulas.
Method 02

Name the identity job

Applies toSimplifies and exact values

The move

Say expand / compress / double / evaluate before writing.

Why it saves time

Ends random formula shopping.

Worked idea

sin75° → expand via 45+30.
Method 03

General solution template

Applies toTrig equations on ℝ

The move

Find α, stamp sin/cos/tan general form, write n∈ℤ.

Why it saves time

Template is half the scored answer.

Worked idea

sinx=1/2 → x=nπ+(−1)ⁿ(π/6).
Method 04

Principal inverse lock

Applies toInverse values and identities

The move

Write ranges before evaluating or adding.

Why it saves time

Kills wrong-branch MC options.

Worked idea

cos⁻¹(−1/2)=2π/3 not −2π/3.
Method 05

Extraneous check

Applies toSquared equations / Weierstrass

The move

Plug every candidate back into the original.

Why it saves time

Ghost roots are designed distractors.

Method 06

Triangle case name

Applies toSOT / area joins

The move

Sketch and name SAS/ASA/SSS/SSA before choosing a law.

Why it saves time

Chooses Cosines vs Sines in seconds.

Worked idea

Two sides + included angle → (1/2)ab sinC or Cosines.

When not to force it

Pure inverse/equation stems — do not force a triangle sketch.

Exam voice

Worked patterns (copy this thinking)

Practise writing these rough-sheet flows until they feel automatic.

Pattern 01

Exact value via compound angles

Exam signalsin/cos/tan of 15°/75° or A±B

Thinking order

  1. Split angle
  2. Expand
  3. Simplify

Rough sheet skeleton

  • 45±30
  • Expand
  • Value

Final check

sin/cos ∈ [−1,1].

Pattern 02

General solution from α

Exam signalSolve … for x∈ℝ

Thinking order

  1. Reduce
  2. α
  3. Template
  4. n∈ℤ

Rough sheet skeleton

  • α=…
  • General form
  • Check

Final check

n=0,1 satisfy original.

Pattern 03

Inverse principal value

Exam signalFind sin⁻¹/cos⁻¹/tan⁻¹ …

Thinking order

  1. Range
  2. Value
  3. Reject other branch

Rough sheet skeleton

  • Range
  • Value

Final check

Answer inside principal interval.

Pattern 04

SAS area / side

Exam signalTwo sides + included angle

Thinking order

  1. Case SAS
  2. (1/2)ab sinC or Cosines
  3. Check

Rough sheet skeleton

  • Case
  • Tool
  • Answer

Final check

Δ>0; sides positive.

Instant recall

Mental bank

Keep these cold — no derivation during the 60–90s window.

  • sin(A±B), cos(A±B), tan(A±B)
  • sin2A, cos2A (three forms), power-reduce
  • sinx=k → nπ+(−1)ⁿα; cos → 2nπ±α; tan → nπ+α
  • sin⁻¹∈[−π/2,π/2]; cos⁻¹∈[0,π]; tan⁻¹∈(−π/2,π/2)
  • sin⁻¹x+cos⁻¹x=π/2 (|x|≤1)
  • a/sinA=2R; Δ=(1/2)bc sinA
  • 15°=45°−30°; factor before expand-all
  • Route page first: Formulas / Equations / Inverse / Triangles
  • Name the identity job before shopping formulae
  • General solution template with correct period
  • Principal inverse range before value
  • Formulas page owns A±B and factor; Equations owns templates; Inverse owns ranges; Triangles owns Sines/Cosines.
  • SOT is join insurance — revise weekly, not as a promise of direct Advanced weight.
Inside Trig

Sub-portion playbooks (detailed)

Each family: explanation, paper look, template, formulae, worked mini, traps, drill.

01 · Foundation + Calculus joins

Sin / Cos / Tan formulas & applications

60s triggerSimplify / exact value → name expand/compress/double → identity → evaluate.

What this family really tests

This is the engine room for every later Trig page and for Calculus joins. Equations feel easy only after A±B, double-angle, and factor moves are automatic. Advanced rarely asks ‘state the formula’ — it asks you to name the job (expand / compress / double / evaluate) in 10s and finish an exact value or feed a limit/integral.

How it looks in the paper

  • Exact sin75° / cos15° style SC
  • Simplify before a limit or ∫sin²
  • Identity T/F with close distractors

Template (write on rough)

  1. Name the identity job.
  2. Apply A±B, double, or factor.
  3. Finish value or feed Calculus.
  4. Tiny-angle test on MC identities.

Must automate

  • A±B and tan(A±B)
  • Double / power-reduce
  • Factor and product→sum
  • π/12 via 45±30

Formula sheet for this family

  • sin(A±B), cos(A±B), tan(A±B)
  • sin2A, cos2A, sin²A=(1−cos2A)/2
  • sinC+sinD=2sin((C+D)/2)cos((C−D)/2)

Traps

  • Wrong cos(A±B) signs
  • Expand-all when factor compresses
  • Ignoring tan undefined

Why marks leak

  • Wrong cos(A±B) sign under pressure
  • Expanding a sum that should have been factored

Drill

Daily: 2 exact values + 1 factor simplify.

Open full Sin / Cos / Tan formulas & applications notes →

Mini worked example

sin75° → expand 45+30 → (√6+√2)/4. On rough sheet write only: ‘expand’, then the two-term line — no formula shopping.
02 · Official Advanced core

Trigonometric equations (general solutions)

60s triggerEquation on ℝ → reduce → α → sin/cos/tan template → n∈ℤ.

What this family really tests

Equation marks are template marks. Finding α is only half the answer — Advanced expects the full general form with n∈ℤ, and counts in an interval when asked. Squaring or Weierstrass creates ghost roots that sit in options; plug-check is part of the scored method, not optional hygiene.

How it looks in the paper

  • Solve … for x∈ℝ (general solution as the answer)
  • How many solutions in [0,2π) / (0,π]
  • Reduce a cos²/sin mix to one function, then template

Template (write on rough)

  1. Isolate one function (or R-method).
  2. Find α.
  3. Stamp general form.
  4. Plug-check if squared.

Must automate

  • Three general templates
  • R-method for a sin+b cos
  • Extraneous root check
  • Count solutions in an interval

Formula sheet for this family

  • sinθ=sinα ⇒ nπ+(−1)ⁿα
  • cosθ=cosα ⇒ 2nπ±α
  • tanθ=tanα ⇒ nπ+α

Traps

  • Only [0,2π] when ℝ asked
  • Wrong period
  • Missing ± for cos

Why marks leak

  • Shipping only [0,2π] when ℝ was asked
  • Missing a branch after reduction (e.g. only sin=1/2, forgetting sin=−1)

Drill

Daily: 2 full general solutions written out.

Open full Trigonometric equations (general solutions) notes →

Mini worked example

sin x = 1/2 → α=π/6 → x = nπ+(−1)ⁿ(π/6). Check n=0,1 give π/6 and 5π/6 — then stop expanding.
03 · Official Advanced — high trap density

Inverse trigonometry (principal values)

60s triggerInverse ask → write principal range → evaluate → identity only if domain-safe.

What this family really tests

Inverse Advanced is principal-range discipline — the same habit as Arg. Options are built from the other branch (e.g. −2π/3 vs 2π/3 for cos⁻¹). Write the range before arithmetic; use a right triangle for compositions; gate arctan sums with ab < 1 before naming π/4-style answers.

How it looks in the paper

  • Principal value of sin⁻¹/cos⁻¹/tan⁻¹ (often with negatives)
  • Simplify sin(cos⁻¹ x) / tan(sin⁻¹ x)
  • Sum of two arctans with an ab-product trap

Template (write on rough)

  1. Range lock.
  2. Principal value.
  3. Composition triangle if needed.
  4. Reject non-principal options.

Must automate

  • Three principal ranges
  • sin⁻¹x+cos⁻¹x=π/2
  • Triangle for sin(cos⁻¹x)
  • Arctan ab>1 gate

Formula sheet for this family

  • sin⁻¹∈[−π/2,π/2]; cos⁻¹∈[0,π]; tan⁻¹∈(−π/2,π/2)
  • sin⁻¹x+cos⁻¹x=π/2
  • sin(cos⁻¹x)=√(1−x²)

Traps

  • Negative cos⁻¹ values
  • π/2 identity outside |x|≤1
  • Arctan sum without ab check

Why marks leak

  • Picking the non-principal branch that ‘looks familiar’
  • Using sin⁻¹x+cos⁻¹x=π/2 outside |x|≤1

Drill

Daily: 3 principal values (include negatives).

Open full Inverse trigonometry (principal values) notes →

Mini worked example

cos⁻¹(−1/2): range [0,π] ⇒ 2π/3 (not −2π/3). Rough sheet: ‘range [0,π]’ first line.
04 · Not a direct Advanced unit — keep for joins

Solution of triangles (join toolkit)

60s triggerTriangle data → sketch + SAS/ASA/SSS/SSA → Sines/Cosines/area.

What this family really tests

Use this page when a stem hides a triangle inside Vectors, Coordinate, or integrals — not as your first Trig revision block every day. The scoring habit is case-name in 10s (SAS/ASA/SSS/SSA), then the matching law. Heron on SAS is a classic time tax.

How it looks in the paper

  • Hidden SAS area inside a Vectors / coordinate stem
  • a/sinA = 2R bridge to find R or a missing side
  • SSA with an ambiguous-case option trap

Template (write on rough)

  1. Sketch and name case.
  2. Pick Law of Sines or Cosines.
  3. Area with matching formula.
  4. SSA ambiguous gate if needed.

Must automate

  • a/sinA=2R
  • Cosine law
  • Δ=(1/2)bc sinA
  • SSA ambiguous check

Formula sheet for this family

  • a/sinA=b/sinB=c/sinC=2R
  • c²=a²+b²−2ab cosC
  • Δ=(1/2)bc sinA

Traps

  • Heron on SAS
  • Ignoring ambiguous SSA
  • Expecting standalone Advanced weight

Why marks leak

  • Skipping SOT entirely, then freezing on a join
  • Wrong law for the given data

Drill

Daily: 1 SAS area + 1 Sines/2R.

Open full Solution of triangles (join toolkit) notes →

Mini worked example

a=5,b=6,∠C=30° → case SAS → Δ=(1/2)·5·6·(1/2)=15/2. Do not open Heron.
Habit

Common practice system

Same system every week — builds transferable speed.

Daily

  • 1 Formulas exact-value or factor.
  • 1 Equations general solution (template written).
  • 1 Inverse principal value with range first.
  • Alt days: 1 SOT SAS/Sines drill.

Weekly

  • 1 mixed Trig set across all four pages.
  • 1 Calculus join needing A±B or power-reduce.
  • 1 Vectors/area join using (1/2)ab sinC.
  • Error log: period / range / case-name misses.

Before every mock

  • Flash: three equation templates + three inverse ranges + 45±30 (4 minutes).
  • Promise: route to the correct Trig topic page before algebra.

Mastery notes

  • Mastery = routing + template automaticity.
  • Tag misses as Route-wrong / Period-wrong / Range-skip / Case-skip / Extraneous.
Full marks gate

100% checklist for Trigonometry

Tick these only when true on timed practice — not when “you know the chapter”.
  • I open the correct Trig topic page within 10s of reading the stem.
  • A±B and factor are automatic on Formulas stems.
  • Every ℝ equation ends with correct general template + n∈ℤ.
  • Every inverse answer checked against principal range.
  • SOT joins start with sketch + case name.
  • Extraneous roots from squaring are plug-checked.
  • I can recite sin/cos/tan general forms in ≤15s.
  • I know SOT is join insurance, not a syllabus promise of direct weight.
  • My weekly error log has anti-trap rules.
  • Accuracy ≥90% on last timed mixed Trig set.
  • Extraneous roots from squaring are plug-checked.
  • Ambiguous SSA is gated before shipping one triangle.
Paper coverage for TrigSyllabus mapTrendsMath · 300 plan
Also study

Other Math note pages

Algebra →Calculus →Coordinate Geometry →Vectors & 3D Geometry →Formulas →Equations →Inverse →Triangles →