This comprehensive guide provides all essential trigonometric formulas aligned with the Class 11 mathematics curriculum. These formulas form the foundation for advanced mathematics and are crucial for board exams, competitive tests like JEE and NEET.
1. Basic Trigonometric Ratios
| Function | Definition | Reciprocal Identity |
|---|---|---|
| sin θ | Opposite/Hypotenuse | cosec θ = 1/sin θ |
| cos θ | Adjacent/Hypotenuse | sec θ = 1/cos θ |
| tan θ | Opposite/Adjacent = sin θ/cos θ | cot θ = 1/tan θ |
| cosec θ | Hypotenuse/Opposite | cosec θ = 1/sin θ |
| sec θ | Hypotenuse/Adjacent | sec θ = 1/cos θ |
| cot θ | Adjacent/Opposite = cos θ/sin θ | cot θ = 1/tan θ |
2. Fundamental Pythagorean Identities
| Identity | Formula |
|---|---|
| Primary Identity | sin²θ + cos²θ = 1 |
| Derived Identity 1 | 1 + tan²θ = sec²θ |
| Derived Identity 2 | 1 + cot²θ = cosec²θ |
Note: These three identities can be derived from each other by dividing the primary identity by sin²θ or cos²θ.
3. Sign Convention in Different Quadrants
| Quadrant | Angle Range | Positive Functions | Memory Aid |
|---|---|---|---|
| I | 0° to 90° | All functions positive | All |
| II | 90° to 180° | sin θ, cosec θ | Silver |
| III | 180° to 270° | tan θ, cot θ | Tea |
| IV | 270° to 360° | cos θ, sec θ | Cups |
4. Trigonometric Values of Standard Angles
| Angle (θ) | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
| cot θ | ∞ | √3 | 1 | 1/√3 | 0 |
| sec θ | 1 | 2/√3 | √2 | 2 | ∞ |
| cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |
5. Complementary Angle Formulas (90° – θ)
| Formula | Result |
|---|---|
| sin(90° – θ) | cos θ |
| cos(90° – θ) | sin θ |
| tan(90° – θ) | cot θ |
| cot(90° – θ) | tan θ |
| sec(90° – θ) | cosec θ |
| cosec(90° – θ) | sec θ |
6. Supplementary Angle Formulas (180° – θ)
| Formula | Result |
|---|---|
| sin(180° – θ) | sin θ |
| cos(180° – θ) | -cos θ |
| tan(180° – θ) | -tan θ |
| cot(180° – θ) | -cot θ |
| sec(180° – θ) | -sec θ |
| cosec(180° – θ) | cosec θ |
7. Angle Addition Formulas (180° + θ)
| Formula | Result |
|---|---|
| sin(180° + θ) | -sin θ |
| cos(180° + θ) | -cos θ |
| tan(180° + θ) | tan θ |
| cot(180° + θ) | cot θ |
8. Angle Formulas (360° – θ) or (-θ)
| Formula | Result |
|---|---|
| sin(360° – θ) or sin(-θ) | -sin θ |
| cos(360° – θ) or cos(-θ) | cos θ |
| tan(360° – θ) or tan(-θ) | -tan θ |
| cot(360° – θ) or cot(-θ) | -cot θ |
9. Sum and Difference Formulas
| Type | Formula |
|---|---|
| sin(A + B) | sin A cos B + cos A sin B |
| sin(A – B) | sin A cos B – cos A sin B |
| cos(A + B) | cos A cos B – sin 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) |
| tan(A – B) | (tan A – tan B)/(1 + tan A tan B) |
| cot(A + B) | (cot A cot B – 1)/(cot B + cot A) |
| cot(A – B) | (cot A cot B + 1)/(cot B – cot A) |
10. Double Angle Formulas
| Function | Formulas |
|---|---|
| sin 2θ | 2 sin θ cos θ |
| cos 2θ | cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ |
| tan 2θ | 2 tan θ/(1 – tan²θ) |
| cot 2θ | (cot²θ – 1)/(2 cot θ) |
11. Triple Angle Formulas
| Function | Formula |
|---|---|
| sin 3θ | 3 sin θ – 4 sin³θ |
| cos 3θ | 4 cos³θ – 3 cos θ |
| tan 3θ | (3 tan θ – tan³θ)/(1 – 3 tan²θ) |
12. Half Angle Formulas
| Function | Formula |
|---|---|
| sin²(θ/2) | (1 – cos θ)/2 |
| cos²(θ/2) | (1 + cos θ)/2 |
| tan²(θ/2) | (1 – cos θ)/(1 + cos θ) |
| sin(θ/2) | ±√[(1 – cos θ)/2] |
| cos(θ/2) | ±√[(1 + cos θ)/2] |
| tan(θ/2) | sin θ/(1 + cos θ) = (1 – cos θ)/sin θ |
13. Product-to-Sum Formulas
| Product | Sum Formula |
|---|---|
| sin A sin B | [cos(A – B) – cos(A + B)]/2 |
| cos A cos B | [cos(A – B) + cos(A + B)]/2 |
| sin A cos B | [sin(A + B) + sin(A – B)]/2 |
| cos A sin B | [sin(A + B) – sin(A – B)]/2 |
14. Sum-to-Product Formulas
| Sum/Difference | Product Formula |
|---|---|
| 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] |
15. Important Trigonometric Limits (for Calculus)
| Limit | Value |
|---|---|
| lim (sin θ)/θ as θ→0 | 1 |
| lim (tan θ)/θ as θ→0 | 1 |
| lim (1 – cos θ)/θ as θ→0 | 0 |
16. Domain and Range of Trigonometric Functions
| Function | Domain | Range |
|---|---|---|
| sin θ | All real numbers (R) | [-1, 1] |
| cos θ | All real numbers (R) | [-1, 1] |
| tan θ | R – {(2n+1)π/2, n ∈ Z} | All real numbers (R) |
| cot θ | R – {nπ, n ∈ Z} | All real numbers (R) |
| sec θ | R – {(2n+1)π/2, n ∈ Z} | (-∞, -1] ∪ [1, ∞) |
| cosec θ | R – {nπ, n ∈ Z} | (-∞, -1] ∪ [1, ∞) |
17. Trigonometric Equations – General Solutions
| Equation | General Solution |
|---|---|
| sin θ = 0 | θ = nπ, where n ∈ Z |
| cos θ = 0 | θ = (2n+1)π/2, where n ∈ Z |
| tan θ = 0 | θ = nπ, where n ∈ Z |
| sin θ = sin α | θ = nπ + (-1)ⁿα, where n ∈ Z |
| cos θ = cos α | θ = 2nπ ± α, where n ∈ Z |
| tan θ = tan α | θ = nπ + α, where n ∈ Z |
| sin²θ = sin²α | θ = nπ ± α, where n ∈ Z |
| cos²θ = cos²α | θ = nπ ± α, where n ∈ Z |
| tan²θ = tan²α | θ = nπ ± α, where n ∈ Z |
18. Inverse Trigonometric Functions – Principal Values
| Function | Domain | Principal Value Range |
|---|---|---|
| sin⁻¹x | [-1, 1] | [-π/2, π/2] |
| cos⁻¹x | [-1, 1] | [0, π] |
| tan⁻¹x | R | (-π/2, π/2) |
| cot⁻¹x | R | (0, π) |
| sec⁻¹x | (-∞, -1] ∪ [1, ∞) | [0, π] – {π/2} |
| cosec⁻¹x | (-∞, -1] ∪ [1, ∞) | [-π/2, π/2] – {0} |
19. Properties of Inverse Trigonometric Functions
| Property | Formula |
|---|---|
| sin⁻¹(-x) | -sin⁻¹x |
| cos⁻¹(-x) | π – cos⁻¹x |
| tan⁻¹(-x) | -tan⁻¹x |
| sin⁻¹x + cos⁻¹x | π/2 |
| tan⁻¹x + cot⁻¹x | π/2 |
| sec⁻¹x + cosec⁻¹x | π/2 |
| tan⁻¹x + tan⁻¹y | tan⁻¹[(x + y)/(1 – xy)] if xy < 1 |
| tan⁻¹x – tan⁻¹y | tan⁻¹[(x – y)/(1 + xy)] |
| 2tan⁻¹x | tan⁻¹[2x/(1 – x²)] if |
Study Tips for Mastering These Formulas
- Practice Strategy: Start with basic identities and progressively move to complex formulas. Regular practice of deriving formulas helps in retention rather than rote memorization.
- Memory Techniques: Use mnemonics like “All Silver Tea Cups” for quadrant signs, and practice deriving sum/difference formulas from the unit circle.
- Application Focus: These formulas are not just for exams they’re fundamental to physics (wave motion, oscillations), engineering, and higher mathematics.
- Common Mistakes to Avoid: Confusing sum and difference formulas, incorrect signs in different quadrants, and forgetting domain restrictions for inverse functions.
- Important Note: This compilation follows CBSE Class 11 and NCERT curriculum standards. All formulas are verified against standard mathematical references and are presented in their most commonly used forms in Indian educational institutions. Students preparing for board exams, JEE, NEET, or other competitive examinations will find this resource complete for their trigonometry requirements.
FAQs on Trigonometry Formulas for Class 11
Q. What are the most important trigonometry formulas for Class 11?
The most important trigonometry formulas for Class 11 include:
- Pythagorean Identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ
- Sum and Difference Formulas: sin(A ± B), cos(A ± B), tan(A ± B)
- Double Angle Formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ – sin²θ
- Compound Angle Formulas: Essential for solving complex problems
- Inverse Trigonometric Functions: Domain and range properties
These formulas form the foundation for Class 11 and 12 mathematics, competitive exams like JEE Main, JEE Advanced, and NEET. Students should focus on understanding derivations rather than memorization for better retention and application.
Q. How many trigonometry formulas are there in Class 11?
Class 11 trigonometry curriculum includes approximately 60-70 core formulas across different categories:
- 6 basic trigonometric ratios and reciprocal identities
- 3 Pythagorean identities
- 24 formulas for allied angles (90°±θ, 180°±θ, 270°±θ, 360°±θ)
- 8 sum and difference formulas
- 4 double angle formulas
- 3 triple angle formulas
- 8 product-to-sum and sum-to-product formulas
- 6 inverse trigonometric function properties
- Multiple general solutions for trigonometric equations
However, many formulas can be derived from basic ones. Students should master the fundamental formulas and understand the derivation process to reduce memorization burden. NCERT textbooks and CBSE syllabus cover all these formulas systematically.
Q. What is the easiest way to memorize trigonometry formulas for Class 11?
Here are proven strategies to memorize trigonometry formulas effectively:
1. Understand, Don’t Memorize: Learn derivations of formulas from basic principles. When you understand how sin(A+B) is derived, you won’t need to memorize it.
2. Use Mnemonics:
- “All Silver Tea Cups” for quadrant signs (All positive, Sin positive, Tan positive, Cos positive)
- “Some People Have Curly Black Hair Through Proper Brushing” for sin = P/H, cos = B/H, tan = P/B
3. Practice Daily: Solve 5-10 problems daily using different formulas to build muscle memory.
4. Create Formula Sheets: Write formulas by hand multiple times. Visual and kinesthetic learning reinforces memory.
5. Group Related Formulas: Study sum/difference formulas together, then double angles, then product-to-sum.
6. Use Flashcards: Digital or physical flashcards with formula on one side and derivation/application on the other.
7. Teach Others: Explaining formulas to classmates strengthens your understanding.
Q. Where can I download trigonometry formulas for Class 11 PDF?
Trigonometry formulas PDFs are available from several reliable educational sources:
Official Sources:
- NCERT Official Website: Free downloadable NCERT textbooks with all formulas
- CBSE Academic Portal: Formula sheets and supplementary materials
- State Education Board Websites: Regional board-specific resources
Our Educational Platform: Shiksha Nation comprehensive PDF formula sheets
Important Tips:
- Verify formulas against NCERT textbooks before relying on any PDF
- Look for PDFs that include derivations, not just formulas
- Choose resources with solved examples and practice problems
- Ensure the PDF follows latest CBSE syllabus guidelines
Caution: Always cross-verify downloaded materials with your textbook, as some online resources may contain errors or outdated information.
Q. Are Class 11 and Class 12 trigonometry formulas the same?
Class 11 and Class 12 trigonometry formulas overlap significantly, but Class 12 builds upon Class 11 foundation:
Class 11 Focuses On:
- Basic trigonometric ratios and identities
- Allied angles and compound angles
- Trigonometric equations (basic level)
- Inverse trigonometric functions and their properties
- Transformation formulas (sum-to-product, product-to-sum)
Class 12 Adds:
- Advanced inverse trigonometric function properties
- Complex trigonometric equations
- Applications in calculus (differentiation and integration of trigonometric functions)
- Trigonometric substitutions in integration
- Properties involving composition of inverse functions
Main Difference: Class 11 establishes foundational formulas and basic problem-solving, while Class 12 applies these formulas in calculus, advanced algebra, and complex problem scenarios.
For Exam Preparation: Students should master Class 11 formulas thoroughly, as they’re prerequisites for Class 12. Most competitive exams like JEE test both Class 11 and 12 concepts together, making complete knowledge essential.
Q. Which trigonometry formulas are most important for JEE and NEET?
For JEE Main, JEE Advanced, and NEET preparation, focus on these high-weightage trigonometry formulas:
Critical for JEE Main & Advanced:
- Sum and Difference Formulas: Used in 40% of trigonometry questions
- Compound Angle Formulas: sin(A±B), cos(A±B), tan(A±B)
- Product-to-Sum Conversions: Essential for integration problems
- General Solutions: sin θ = sin α, cos θ = cos α, tan θ = tan α
- Inverse Trigonometric Properties: tan⁻¹x + tan⁻¹y formulas
- Multiple and Sub-multiple Angles: 2θ, 3θ, θ/2 formulas
Critical for NEET:
- Basic Trigonometric Ratios: For physics problems (vectors, projectile motion)
- Pythagorean Identities: Used in wave optics and oscillations
- Small Angle Approximations: sin θ ≈ θ, cos θ ≈ 1 for θ → 0
- Double Angle Formulas: Applications in physics derivations
Q. How to solve trigonometric equations using Class 11 formulas?
Follow this systematic approach to solve trigonometric equations:
Step 1: Simplify the Equation
- Use Pythagorean identities to convert all functions to sin θ or cos θ
- Example: sec²θ can be replaced with 1 + tan²θ
Step 2: Apply Appropriate Formulas
- Use compound angle formulas for sin(A+B) type expressions
- Apply double/triple angle formulas where needed
- Convert products to sums using product-to-sum formulas
Step 3: Reduce to Standard Form
- Aim for equations like sin θ = k, cos θ = k, or tan θ = k
- Factor quadratic trigonometric equations
Step 4: Find Principal Solutions
- Determine solutions in [0, 2π] or [0°, 360°] range
- Use inverse trigonometric functions carefully
Step 5: Write General Solutions
- sin θ = sin α → θ = nπ + (-1)ⁿα
- cos θ = cos α → θ = 2nπ ± α
- tan θ = tan α → θ = nπ + α
Q. What are the basic trigonometric identities every Class 11 student must know?
Every Class 11 student must master these fundamental trigonometric identities:
1. Reciprocal Identities (6 formulas):
- sin θ = 1/cosec θ, cos θ = 1/sec θ, tan θ = 1/cot θ
- cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
2. Quotient Identities (2 formulas):
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
3. Pythagorean Identities (3 formulas):
- sin²θ + cos²θ = 1 (most fundamental)
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
4. Even-Odd Identities:
- sin(-θ) = -sin θ (odd function)
- cos(-θ) = cos θ (even function)
- tan(-θ) = -tan θ (odd function)
5. Cofunction Identities:
- sin(90° – θ) = cos θ
- cos(90° – θ) = sin θ
- tan(90° – θ) = cot θ
Why These Matter: These identities are the building blocks for all advanced formulas. They appear in every trigonometry problem, whether in algebra, calculus, or physics applications. Master these first before moving to compound angles and transformations.
Learning Sequence: Start with Pythagorean identities → reciprocal identities → quotient identities → cofunction identities. This progression builds natural understanding.