Trigonometry — JEE Main & Advanced Notes

Master identities, transformations, equations and triangle applications with quadrant discipline.

Material signal: Mapped from log, modulus, trigonometry and equation revision sets.

Priority: High Yield. Treat this as a foundation chapter inside the JEE Mathematics ladder.

• Quadrant ASTC rule: All (sin,cos,tan positive) in Q1; Sine positive in Q2; Tangent positive in Q3; Cosine positive in Q4. When an identity involves a square root (e.g. cos(x/2) = ±√((1+cosx)/2)), the sign is determined by the quadrant of x/2, not x. Always ask 'which quadrant does the angle fall in?' before assigning a sign.
• Pythagorean identities as algebraic tools: sin²x+cos²x=1 is used constantly — not only to verify but to substitute. If a problem has sin²x and cos²x mixed, replace one with 1−(other)². The identities 1+tan²x=sec²x and 1+cot²x=cosec²x appear in integrals and equations involving sec and cosec. Express everything in terms of sin and cos first when unsure.
• Sum-to-product and product-to-sum: These are the key to proving identities and simplifying sums. sinA+sinB splits into a product, making factoring possible. In reverse, 2sinA cosB becomes a sum — useful in integration. Recognise which direction is needed: if you see a sum of trig functions, try sum-to-product; if you see a product, try product-to-sum.
• Multiple angle formulas — when they appear: sin2A, cos2A and tan2A appear whenever a problem involves both a function and its double angle, or when the angle is written as 2×something. The cos2A formula has three equivalent forms — use cos2A=1−2sin²A when only sinA is given; use cos2A=2cos²A−1 when only cosA is given; use the difference form when both are available.
• Solving trig equations — complete general solution: First find the principal value from the inverse trig function (calculator-style). Then apply the correct general solution template depending on which function is involved (sin, cos or tan). If sinθ=k and cosθ=m together, use sin²+cos²=1 to check consistency and find specific solutions. For equations like sin2x=cosx, convert to a single function or use double-angle to factor.
• Sine rule and cosine rule — when to use each: Use the sine rule when two angles and one side are known (AAS, ASA) or two sides and an angle opposite one of them (SSA — the ambiguous case). Use the cosine rule when two sides and the included angle are known (SAS) or all three sides (SSS). The cosine rule gives the third side or the angle; the sine rule then gives the remaining parts faster.
• Range of trig expressions: For a·sinx + b·cosx, the range is [−√(a²+b²), √(a²+b²)]. Write it as R·sin(x+φ) where R=√(a²+b²) and tanφ=b/a. For finding max/min of more complex expressions, substitute the Pythagorean identity to reduce to one variable, then differentiate or complete the square.
• Inverse trig functions — range restrictions: sin⁻¹ has range [−π/2, π/2]; cos⁻¹ has range [0, π]; tan⁻¹ has range (−π/2, π/2). These restricted ranges ensure the inverse is a function. Key identities: sin⁻¹x + cos⁻¹x = π/2; sin⁻¹(−x) = −sin⁻¹(x); cos⁻¹(−x) = π−cos⁻¹(x).

• sin²x + cos²x = 1; 1 + tan²x = sec²x; 1 + cot²x = cosec²x
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• sin2A = 2sinA cosA = 2tanA/(1+tan²A); cos2A = cos²A−sin²A = (1−tan²A)/(1+tan²A)
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• tan2A = 2tanA/(1−tan²A)
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• sin3A = 3sinA−4sin³A; cos3A = 4cos³A−3cosA
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• sinA + sinB = 2 sin((A+B)/2) cos((A−B)/2); sinA − sinB = 2 cos((A+B)/2) sin((A−B)/2)
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• cosA + cosB = 2 cos((A+B)/2) cos((A−B)/2); cosA − cosB = −2 sin((A+B)/2) sin((A−B)/2)
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• 2sinA cosB = sin(A+B) + sin(A−B); 2cosA cosB = cos(A+B) + cos(A−B); 2sinA sinB = cos(A−B) − cos(A+B)
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• Sine rule: a/sinA = b/sinB = c/sinC = 2R
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• Cosine rule: a² = b²+c²−2bc cosA
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• Area of triangle = (1/2)ab sinC = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• General solution of sinθ = sinα: θ = nπ + (−1)ⁿα, n∈ℤ
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• General solution of cosθ = cosα: θ = 2nπ ± α, n∈ℤ
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• General solution of tanθ = tanα: θ = nπ + α, n∈ℤ
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.
• Half-angle: sin²(x/2) = (1−cosx)/2; cos²(x/2) = (1+cosx)/2
  Hint: do not memorise this in isolation; connect it to the definition or diagram that produces it.

• Dropping the general solution: solving sinx=1/2 and writing only x=π/6 is incomplete. The full answer is x = nπ+(−1)ⁿ(π/6). In an interval question, find all n that keep x within the given range.
• Sign errors with half-angle square roots: cos(x/2) = ±√((1+cosx)/2). The sign depends on the quadrant of x/2, not x. For x∈(π,3π/2), x/2∈(π/2,3π/4) which is Q2, so cos(x/2) < 0. Always determine the quadrant of the half-angle explicitly.
• Confusing an identity (true for all x) with an equation (true at specific x): sin2x=2sinxcosx is always true. sinx=cosx is true only at specific x. Proving an identity requires showing both sides are equal algebraically, not solving for x.
• Using the sine rule in the ambiguous case without checking: given two sides and a non-included angle (SSA), there may be 0, 1 or 2 valid triangles. Always check sin(angle) ≤ 1 and whether two angles are geometrically possible.
• Forgetting to use radian mode for calculus-adjacent problems: in problems mixing derivatives of trig functions with trig values, angles must be in radians. sin(90°)=1 but the derivative identity d/dx(sinx)=cosx is valid only when x is in radians.

For JEE Main, aim for fast recognition and clean substitution. Finish the first pass of Trigonometry questions in 60–90 seconds each. Prioritise standard formulas, short sign/domain checks and option elimination only after the setup is correct.

For JEE Advanced, expect combined conditions, hidden domains, multi-correct traps and integer-style answers. Build the solution from definitions, not memorised tricks. When a parameter appears, solve the general case and then filter using restrictions.

• Standard values: sin0=0, sin30=1/2, sin45=1/√2, sin60=√3/2, sin90=1. Cosine is the reverse. Tangent: 0,1/√3,1,√3,undefined.
• To prove an identity: start from the more complex side and simplify toward the simpler. Convert everything to sin and cos if stuck. Look for Pythagorean substitution opportunities.
• General solution template: sinθ=sinα → θ=nπ+(−1)ⁿα. cosθ=cosα → θ=2nπ±α. tanθ=tanα → θ=nπ+α.
• For a·sinx+b·cosx expressions: write as R·sin(x+φ) with R=√(a²+b²), then max=R, min=−R.
• In triangles: sine rule for AAS/ASA/SSA; cosine rule for SAS/SSS. Area = (1/2)ab sinC always works.