Math notesTrigonometryFormulas
Independent topic page
Sin / 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.
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.
04. Full marks: write the identity name on rough sheet, substitute numbers carefully, and check domain (tan undefined, cos≠0 in divisions).
05. Advanced lane: name the job — expand, compress, double, or evaluate — then pick one identity family. Random formula shopping burns the 90s window.
06. Exact values at π/12, π/8 style angles: build from 45±30 or half-angle; do not invent decimals under Advanced pressure.
Official · 2026
Official syllabus points for Trig Formulas & Applications only — not the full Trigonometry portion map. Portion hub: Trigonometry · Trigonometry analysis.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusCompound/multiple-angle identities and graph/periodicity — not inverse/general solutions alone.
Syllabus points
2020–2026 paper weight
Topic match inside parent portion Trigonometry (10.3% of Math tags). Stats below are portion backdrop; matched subtopics show where Trig Formulas & Applications sits.
Portion-wide type/difficulty mix lives on the Trigonometry analysis — not repeated here so this page stays on Formulas.
01. Dominant subtopics: Trigonometric Equations (18), Inverse Trigonometry (6), Triangle/Trigonometry (1).
Cross-topic stems
Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Trig Formulas & Applications — practise the knowledge bridge, not only the single-chapter drill.
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.
The stem hides trig solving inside an exponential envelope; students often integrate before finding exact limits.
Drill ‘cancel common positive factor → trig roots → area integral’ as one reflex chain.
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.
Classic Advanced join: geometry conditions feeding calculus numerics in the same paper.
Any week you revise conics, end with one area-between-curves NUM.
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.
Heavy NUM opening stresses accuracy before you are warm.
Start Math mocks with a 10-minute trig warm-up when the paper is NUM-heavy.
Why methods matter
Formulas are a job list: expand, compress, double, evaluate. These methods stop random identity shopping and make π/12 and factor moves automatic.
Stops random formula shopping — expand/compress/double/evaluate becomes automatic.
Do thisSay expand / compress / double / evaluate before writing algebra.
Stops random formula shopping under pressure.
sin75°=sin(45+30) → expand job → exact value in one line.
Exact 15°/75° values become bankable instead of guesswork.
Do thisFor 15°/75°, always use 45°±30° compound — never invent new angles.
Bankable exact values appear every few papers and in joins.
cos15°=cos(45−30)=(√6+√2)/4.
Sums that look impossible compress in one line.
Do thisSums of sin/cos → factor formulae first; only expand if factor fails.
Compresses expressions that look impossible.
sin3x+sin x = 2sin2x cos x — then solve or simplify.
False MC identities die in seconds — toppers use this constantly.
Do thisOn opaque MC identities, plug x=π/6 or π/4 to kill false options.
Faster than full proof when options are absolute claims.
If LHS≠RHS at π/6, the identity is false — tick and move.
Processed examples
Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.
Find the exact value of sin 75°.
Name the job ‘expand’ — 75°=45°+30° is automatic, not invention.
Model chosensin(45+30)=sin45 cos30+cos45 sin30.
Answer(√6+√2)/4
Takeaway: π/12 family always via 45±30.
Simplify sin 3x + sin x.
Factor before expand-all — compress is the job.
Model chosensinC+sinD=2sin((C+D)/2)cos((C−D)/2).
Answer2 sin 2x cos x
Takeaway: Sums of sin → factor first; write C,D before the formula.
Evaluate exactly cos 20° · cos 40° · cos 80° (no decimals).
Product collapses via repeated double-angle — classic Advanced exact-value.
Model chosensin 2θ = 2 sin θ cos θ repeatedly → sin 8θ identity.
Answer1/8
Takeaway: For cos θ cos 2θ cos 4θ products, build from sin 8θ.
Reusable models
Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.
Looks likesin/cos/tan of 15°, 75°, or compound angles
Fast modelSplit into 45±30 (or other known) → expand A±B → simplify.
One expand beats inventing new half-angle paths.
Do not start from Weierstrass for a pure exact-value ask.
Looks likeSimplify sinC±sinD; convert products before integrate/limit
Fast modelName compress vs product→sum; apply one identity; stop.
Compression removes algebra that invites sign errors.
Do not expand every term into powers of sin/cos first.
Looks likesin², cos², sin2A forms in simplify or Calculus joins
Fast modelsin²A=(1−cos2A)/2; spot 2A forms early.
Standard Calculus feeders finish in seconds.
Do not integrate sin² by parts when power-reduce works.
Looks likeProve identity; find condition so identity holds; MATCH left↔right
Fast modelMove all to one side → factor / express in one function → verify ≡0.
One-side discipline stops circular ‘rewrites’.
Do not assume the identity while ‘simplifying’ both sides independently forever.
Hall timing
Spot whether you need expand (A±B), compress (factor), or double/half. Note if tan is involved (undefined checks).
Standard angles: 0, π/6, π/4, π/3, π/2 and π/12 via 15°=45°−30°.
If joining Calculus, stop once the expression is a known standard form.
MC: kill options with a tiny numeric angle (e.x. x=π/6) when identities look opaque.
After the model is locked
sin(A±B)=sinA cosB±cosA sinB; cos(A±B)=cosA cosB∓sinA sinB
Compound expand/reduce
tan(A±B)=(tanA±tanB)/(1∓tanA tanB)
Tangent compound; check 1∓tanA tanB≠0
sin2A=2sinA cosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A)
Multiple-angle compress/expand
sin(A/2)=±√((1−cosA)/2); cos(A/2)=±√((1+cosA)/2); tan(A/2)=sinA/(1+cosA)
Half-angle; sign from quadrant
sinC+sinD=2sin((C+D)/2)cos((C−D)/2); cosC+cosD=2cos((C+D)/2)cos((C−D)/2)
Sum of sines/cosines
2sinA cosB=sin(A+B)+sin(A−B); 2cosA cosB=cos(A+B)+cos(A−B)
Products before integrate/limit
sin²A=(1−cos2A)/2; cos²A=(1+cos2A)/2
Integrals and averages
sin(x+2π)=sin x; cos(x+2π)=cos x; tan(x+π)=tan x
Graphs, range, general solutions setup
Make it automatic