Trigonometry

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Trigonometry — Key Facts for JEE Main Six basic ratios: sin θ, cos θ, tan θ, cosec θ, sec θ, cot θ Reciprocal identities: sin θ = 1/cosec θ, cos θ = 1/sec θ, tan θ = sin θ/cos θ Pythagorean: sin²θ + cos²θ = 1; sec²θ = 1 + tan²θ; cosec²θ = 1 + cot²θ Signs in quadrants: All S (sin positive in QII), Stop C (tan positive in QIII), Crab (cos positive in QIV) — or use ASTC rule ⚡ Exam tip: JEE Main tests trig identities, equations, and inverse trig — memorise the standard transformations!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Trigonometry — JEE Main Study Guide

Compound angle formulas:

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
• cot(A ± B) = (cot A cot B ∓ 1)/(cot B ± cot A)

Double angle formulas:

• sin 2θ = 2 sin θ cos θ = 2 tan θ/(1 + tan²θ)
• cos 2θ = cos²θ − sin²θ = 1 − 2 sin²θ = 2 cos²θ − 1 = (1 − tan²θ)/(1 + tan²θ)
• tan 2θ = 2 tan θ/(1 − tan²θ)
• cos 2θ = (cot²θ − 1)/(cot²θ + 1)

Triple angle formulas:

• sin 3θ = 3 sin θ − 4 sin³θ
• cos 3θ = 4 cos³θ − 3 cos θ
• tan 3θ = (3 tan θ − tan³θ)/(1 − 3 tan²θ)

Sum-to-product identities:

• sin C + sin D = 2 sin[(C+D)/2] cos[(C−D)/2]
• sin C − sin D = 2 cos[(C+D)/2] sin[(C−D)/2]
• cos C + cos D = 2 cos[(C+D)/2] cos[(C−D)/2]
• cos C − cos D = −2 sin[(C+D)/2] sin[(C−D)/2]

Product-to-sum (useful for integration):

• sin A cos B = ½[sin(A+B) + sin(A−B)]
• cos A cos B = ½[cos(A+B) + cos(A−B)]
• sin A sin B = ½[cos(A−B) − cos(A+B)]

General solution of trig equations:

• sin θ = sin α → θ = nπ + (−1)^n α
• cos θ = cos α → θ = 2nπ ± α
• tan θ = tan α → θ = nπ + α

Radians and degrees: π rad = 180°; conversion: multiply by π/180 to convert degrees to radians

• Key formula: sin²θ + cos²θ = 1; sin(A+B) = sin A cos B + cos A sin B
• Common trap: tan 90° is undefined, not infinity — always be careful about domain restrictions
• Exam weight: 1–2 questions per year (4–8 marks); forms foundation for inverse trig and calculus

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Trigonometry — Comprehensive JEE Main Notes

Maximum-minimum of trig expressions: For a sin x + b cos x: maximum = √(a² + b²), minimum = −√(a² + b²) Achieved at angle φ where tan φ = b/a

R format: a sin x + b cos x = R sin(x + α) where R = √(a²+b²) and α = tan⁻¹(b/a)

Solving complex trig equations:

• Method: use identities to reduce to standard form
• Always check domain restrictions (tan undefined at π/2 + nπ; sec undefined at π/2 + nπ; cosec undefined at nπ)

Conditional identities (when A + B + C = π): For angles of a triangle (A + B + C = π):

• sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
• cos 2A + cos 2B + cos 2C = −1 − 4 cos A cos B cos C
• sin A + sin B + sin C = 4 cos(A/2) cos(B/2) sin(C/2)
• cos A + cos B + cos C = 1 + 4 sin(A/2) sin(B/2) cos(C/2)

Weierstrass substitution (t = tan(x/2)):

• sin x = 2t/(1+t²)
• cos x = (1−t²)/(1+t²)
• tan x = 2t/(1−t²) Useful for integration and solving certain equations

Homogeneous equations: Divide by cos^n or sin^n to convert to tan or cot form Example: sin θ + cos θ = √2 cos(θ − π/4) is not homogeneous For homogeneous degree n: treat like polynomial in tan(θ/2)

T-formula method for equations: For sin x + cos x = k type, use R sin(x + α) or R cos(x − α) For sin x + √3 cos x = √3, find R = √(1+3) = 2, find angle α where sin α = 1/2, cos α = √3/2 → α = π/6 So: sin x + √3 cos x = 2 sin(x + π/3)

Elimination problems: Given two equations in A and B, eliminate one angle Common trick: use sin² + cos² = 1 or use compound angle formulas

Important ranges to remember:

• −1 ≤ sin x ≤ 1, −1 ≤ cos x ≤ 1
• sin⁻¹: [−π/2, π/2]; cos⁻¹: [0, π]; tan⁻¹: (−π/2, π/2)
• sec x ≥ 1 or sec x ≤ −1; cosec x ≥ 1 or cosec x ≤ −1

Transformation cascade: When solving expressions like sin 4x in terms of sin 2x and cos 2x: sin 4x = 2 sin 2x cos 2x And cos 2x = 1 − 2 sin²x → can express in sin x alone

Half-angle formulas:

• sin(θ/2) = ±√[(1 − cos θ)/2]
• cos(θ/2) = ±√[(1 + cos θ)/2]
• tan(θ/2) = ±√[(1 − cos θ)/(1 + cos θ)] = (1 − cos θ)/sin θ = sin θ/(1 + cos θ)

Nth roots of unity in trig:

• cos(2πk/n) + i sin(2πk/n) for k = 0, 1, …, n−1

• Sum of all nth roots of unity = 0

• Remember: ASTC for signs (+ quadrant I, All in II, S (only sin positive), T (only tan positive) in III, C (only cos positive) in IV); sin² + cos² = 1 always; general solutions: sin θ = sin α → θ = nπ + (−1)^nα; cos θ = cos α → θ = 2nπ ± α; tan θ = tan α → θ = nπ + α

• Previous years: “Find general solution of 2 sin²x + sin x − 1 = 0” [2023]; “Prove that tan 20° tan 40° tan 60° tan 80° = 3” [2024]; “Find max value of 3 sin x + 4 cos x” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• Trigonometry: 1–2 questions per year, 4–8 marks
• Common patterns: solving trig equations, proving identities, finding max/min values, compound angle formulas
• Weight: high frequency in Physics too (projectile motion, SHM, etc.)

💡 Pro Tips

• For general solution, always include ”+ 2nπ” or ”+ nπ” as appropriate — missing the periodic part loses marks
• tan(A+B+C) formula when A+B+C = π gets used in triangle problems
• For max/min of a sin x + b cos x, use R sin(x + φ) method — R = √(a²+b²)
• Half-angle formulas: sign depends on which quadrant the angle lies in
• When eliminating between two trig equations, the sin² + cos² = 1 identity is most powerful
• Many trig equations in JEE Main reduce to quadratic in tan(x/2) using Weierstrass substitution

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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