Trigonometric Functions Class 11 Revision Notes JEE NEET — Grandmaster Guide

1. ⚡ Formula Bank
2. 🪤 The 5 Mistakes That Cost Marks
3. ✏️ 3 Solved PYQs
4. 🧠 The One Thing Most Students Get Wrong
5. 👁️ Ayush's Note
6. 🔁 Last 5 Minutes Box
7. 📝 Practice MCQs

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⚡ Formula Bank

Fundamental Identities & Quadrant Rules

• Reciprocal Identities:

  • sin θ = 1/cosec θ
  • cos θ = 1/sec θ
  • tan θ = 1/cot θ
  • tan θ = sin θ/cos θ
  • cot θ = cos θ/sin θ

• Pythagorean Identities:

  • sin² θ + cos² θ = 1
  • 1 + tan² θ = sec² θ
  • 1 + cot² θ = cosec² θ

• ASTC Quadrant Rule (Signage):

  • Quadrant I (0 to π/2): All functions are positive.
  • Quadrant II (π/2 to π): sin and cosec are positive; others are negative.
  • Quadrant III (π to 3π/2): tan and cot are positive; others are negative.
  • Quadrant IV (3π/2 to 2π): cos and sec are positive; others are negative.

• Allied Angle Transformations:

  • sin(π/2 - θ) = cos θ
  • cos(π/2 - θ) = sin θ
  • sin(π - θ) = sin θ (Quadrant II)
  • cos(π - θ) = -cos θ (Quadrant II)
  • sin(π + θ) = -sin θ (Quadrant III)
  • cos(π + θ) = -cos θ (Quadrant III)
  • sin(2π - θ) = -sin θ (Quadrant IV)
  • cos(2π - θ) = cos θ (Quadrant IV)

• Even/Odd Identities:

  • sin(-θ) = -sin θ
  • cos(-θ) = cos θ
  • tan(-θ) = -tan θ

Angle Conversions & Periodicity

• Degree to Radian: Radian = Degree × (π/180)
• Radian to Degree: Degree = Radian × (180/π)
• Arc Length: s = rθ (θ in radians)
• Periodicity:
  • sin, cos, sec, cosec: Period = 2π
  • tan, cot: Period = π
  • sin(ax + b): Period = 2π/|a|
  • tan (ax + b): Period = π/|a|

Compound Angle Formulas (Sum & Difference)

• Sine: sin(α ± β) = sin α cos β ± cos α sin β
• Cosine: cos(α ± β) = cos α cos β ∓ sin α sin β (Note the sign flip)
• Tangent: tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α tan β)
• Cotangent: cot(α ± β) = (cot α cot β ∓ 1) / (cot β ± cot α)

Multiple & Sub-multiple Angle Formulas

• Double Angle (2θ):

  • sin 2θ = 2 sin θ cos θ = 2 tan θ / (1 + tan² θ)
  • cos 2θ = cos² θ - sin² θ = 2 cos² θ - 1 = 1 - 2 sin² θ
  • tan 2θ = 2 tan θ / (1 - tan² θ)

• Triple Angle (3θ):

  • sin 3θ = 3 sin θ - 4 sin³ θ
  • cos 3θ = 4 cos³ θ - 3 cos θ
  • tan 3θ = (3 tan θ - tan³ θ) / (1 - 3 tan² θ)

• Half Angle (θ/2):

  • sin²(θ/2) = (1 - cos θ) / 2
  • cos²(θ/2) = (1 + cos θ) / 2
  • tan(θ/2) = (1 - cos θ) / sin θ = sin θ / (1 + cos θ)

Transformation Formulas (Product ↔ Sum)

• Product to Sum:

  • 2 sin α cos β = sin(α + β) + sin(α - β)
  • 2 cos α sin β = sin(α + β) - sin(α - β)
  • 2 cos α cos β = cos(α + β) + cos(α - β)
  • 2 sin α sin β = cos(α - β) - cos(α + β)

• Sum to Product:

  • 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)

Maximum & Minimum Values

• f(θ) = a sin θ + b cos θ:
  • Max = √(a² + b²)
  • Min = -√(a² + b²)
• f(θ) = a sin² θ + b cos² θ:
  • Max = max(a, b)
  • Min = min(a, b)

General Solutions

• sin θ = sin α: θ = nπ + (-1)ⁿ α
• cos θ = cos α: θ = 2nπ ± α
• tan θ = tan α: θ = nπ + α

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🪤 The 5 Mistakes That Cost Marks

• Mistake 1 — The Inverse Cosine Sign Trap:

  • 🔴 What students write: cos⁻¹(-x) = -cos⁻¹(x)
  • ✅ What examiners expect: cos⁻¹(-x) = π - cos⁻¹(x)
  • 💸 Marks lost: 1 mark
  • 🔧 The fix: The range of cos⁻¹(x) is [0, π]. It can NEVER be negative. If the argument is negative, you must subtract the principal value from π.

• Mistake 2 — The Squaring Ghost (Extraneous Solutions):

  • 🔴 What students write: Squaring both sides of sin θ + cos θ = 1 to get sin 2θ = 0, then accepting all solutions like θ = π.
  • ✅ What examiners expect: Always plug your final θ values back into the original equation to verify. sin(π) + cos(π) = -1, which does not equal 1. Reject θ = π.
  • 💸 Marks lost: 2 marks
  • 🔧 The fix: Squaring creates "ghost" solutions. Always verify the results in the non-squared original equation.

• Mistake 3 — The Periodicity Oversight:

  • 🔴 What students write: Period of tan(3θ) is 2π/3.
  • ✅ What examiners expect: Period of tan(kθ) is π/k. For tan(3θ), the period is π/3.
  • 💸 Marks lost: 1 mark
  • 🔧 The fix: sin and cos are 2π-based; tan and cot are π-based. Divide the base period by the coefficient of θ.

• Mistake 4 — The Half-Angle Sign Flip:

  • 🔴 What students write: sin² θ = (1 + cos 2θ)/2
  • ✅ What examiners expect: sin² θ = (1 - cos 2θ)/2 and cos² θ = (1 + cos 2θ)/2.
  • 💸 Marks lost: 2 marks
  • 🔧 The fix: Remember "Cos is Positive". The identity with the "+" sign belongs to cos² θ.

• Mistake 5 — The ASTC Quadrant Blindness:

  • 🔴 What students write: tan(π + θ) = -tan θ
  • ✅ What examiners expect: tan(π + θ) = tan θ (since tan is positive in the 3rd quadrant).
  • 💸 Marks lost: 1 mark
  • 🔧 The fix: Use the ASTC (All-Silver-Tea-Cups) rule. Identify the quadrant first, then determine the sign.

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✏️ 3 Solved PYQs

Q1 (2026 JEE Main): The value of cos(20°) cos(40°) cos(80°) is equal to?

🪤 Trap: Many students attempt to use sum-to-product formulas for every pair. This leads to a massive algebraic expansion that usually results in calculation errors.

🧮 Solution:

1. Identify the pattern: Angles are in GP with ratio 2 (20, 40, 80).
2. Multiply and divide by 2 sin(20°): [2 sin(20°) cos(20°) cos(40°) cos(80°)] / [2 sin(20°)]
3. Use 2 sin θ cos θ = sin 2θ: [sin(40°) cos(40°) cos(80°)] / [2 sin(20°)]
4. Repeat for 40°: [sin(80°) cos(80°)] / [4 sin(20°)]
5. Repeat for 80°: [sin(160°)] / [8 sin(20°)]
6. Since sin(160°) = sin(180° - 20°) = sin(20°), the expression is sin(20°) / [8 sin(20°)] = 1/8.

Final Answer: 1/8

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Q2 (2026 NEET): The maximum value of f(x) = 5 sin(x) - 12 cos(x) + 7 is?

🪤 Trap: Students often forget to add the constant term (+7) after calculating the amplitude of the trigonometric part.

🧮 Solution:

1. Identify coefficients for the trig part: a = 5, b = -12.
2. Calculate amplitude R = √(a² + b²) = √(5² + (-12)²) = √169 = 13.
3. The max value of (5 sin x - 12 cos x) is 13.
4. Add the constant: Max = 13 + 7 = 20.

Final Answer: 20

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Q3 (2026 JEE Main): If tan(x) + sec(x) = p, then find sin(x) in terms of p.

🪤 Trap: Squaring tan(x) + sec(x) = p immediately creates a mixed term that is hard to isolate.

🧮 Solution:

1. Use sec² x - tan² x = 1 ⇒ (sec x - tan x)(sec x + tan x) = 1.
2. Substitute given: (sec x - tan x) * p = 1 ⇒ sec x - tan x = 1/p.
3. We have: sec x + tan x = p and sec x - tan x = 1/p.
4. Adding gives: 2 sec x = p + 1/p = (p² + 1)/p ⇒ sec x = (p² + 1)/(2p).
5. Subtracting gives: 2 tan x = p - 1/p = (p² - 1)/p ⇒ tan x = (p² - 1)/(2p).
6. sin x = tan x / sec x = [(p² - 1)/(2p)] / [(p² + 1)/(2p)] = (p² - 1)/(p² + 1).

Final Answer: (p² - 1) / (p² + 1)

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🧠 The One Thing Most Students Get Wrong

• The misconception (what 85% believe): The "Cancellation Illusion." Students treat inverse functions as simple algebraic cancellations. They assume sin⁻¹(sin x) = x for all x. If they see sin⁻¹(sin(13π/4)), they blindly write 13π/4.

• The reality (what 99% know): Inverse trigonometric functions return values ONLY within their Principal Value Branch.

  • sin⁻¹(x) range is [-π/2, π/2]
  • cos⁻¹(x) range is [0, π] If the angle is outside these bounds, you must use periodicity or symmetry to map it back.

• The diagnostic question: Evaluate: cos⁻¹(cos(11π/6))

  • If you said 11π/6, you failed the range test (11π/6 > π).
  • Correct approach: cos(11π/6) = cos(2π - π/6) = cos(π/6).
  • Since π/6 is in [0, π], the answer is π/6.

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👁️ Ayush's Note

• 🔮 The Hidden Pattern: Trigonometry is the backbone of Complex Numbers. In many JEE Advanced problems, an identity is just Euler's Formula (e^(iθ) = cos θ + i sin θ) in disguise. Master the link between rotation and trigonometry.
• 🎯 The "Always Check" Rule: The "Domain/Undefined" Trap. Always verify if your θ makes tan θ or sec θ not defined (at π/2 + nπ).
• 📊 PYQ Frequency Intel:
  • 2019: General solutions and Product-to-Sum.
  • 2021: Series summation (Σ cos(nθ)) and Conditional Identities.
  • 2023: Compound angle identities in coordinate geometry.
• ⚡ The 30-Second Shortcut: The "Angle Substitution" Method. For identity questions, don't expand. Plug θ = 45° or θ = 30° (avoiding 0°/90° to prevent "not defined" errors) and check options.

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🔁 Last 5 Minutes Box

⚡ Core Formulas

• sin(A ± B) = sin A cos B ± cos A sin B — Compound angle expansion
• cos(A ± B) = cos A cos B ∓ sin A sin B — Compound angle expansion (Note sign flip)
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B) — Tangent expansion
• sin 2θ = 2 sin θ cos θ — Double angle sine
• cos 2θ = 2 cos² θ - 1 = 1 - 2 sin² θ — Double angle cosine

🧠 Must-Know Facts

• Periodicity: sin, cos, sec, csc have period 2π; tan and cot have period π.
• ASTC Rule: Determines function signs in each quadrant.
• Range: sin θ and cos θ are strictly confined to [-1, 1].

🚫 Never Forget

• ❌ sin(-θ) = sin θ → ✅ sin(-θ) = -sin θ
• ❌ cos(A + B) = cos A cos B + sin A sin B → ✅ cos(A + B) = cos A cos B - sin A sin B

🎯 If you can only remember ONE thing

Always verify the quadrant sign and periodicity before finalizing any trigonometric equation.

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📝 Practice MCQs

1. What is the value of tan(7.5°)?

   • A) √6 - √3 + √2 - 2
   • B) √6 + √3 - √2 - 2
   • C) √6 - √4 + √2 - 1
   • D) √6 - √3 - √2 + 2
   • Correct Answer: A (Use tan(θ/2) formula twice)

2. The period of f(x) = tan(3x + 5) is:

   • A) π
   • B) 2π/3
   • C) π/3
   • D) 3π
   • Correct Answer: C (Period = π/|k|)

3. How many solutions does sin x = x/10 have?

   • A) 3
   • B) 6
   • C) 7
   • D) Infinite
   • Correct Answer: C (Check intersections of y=sin x and y=x/10)

4. The value of cos(1°) cos(2°) cos(3°) ... cos(179°) is:

   • A) 1
   • B) -1
   • C) 0
   • D) 1/2
   • Correct Answer: C (cos 90° = 0)

5. Maximum value of f(x) = 1 / (2 - cos 3x) is:

   • A) 1
   • B) 1/3
   • C) 1/2
   • D) 2
   • Correct Answer: A (Max when denominator is minimum: 2 - 1 = 1)

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This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

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📚 Related Topics

• 📖 Complex Numbers
• 📖 Inverse Trigonometric Functions
• 📖 Limits and Derivatives