Key Trigonometric Identities and Formulas Explained for Students
The concept of Trigonometry Formulas List is essential in mathematics and helps in solving real-world and exam-level problems efficiently.
Understanding Trigonometry Formulas List
A Trigonometry Formulas List refers to a collection of equations that relate the angles and sides of triangles using trigonometric ratios. This concept is widely used in angles, triangles, and geometry, and appears in CBSE and JEE syllabus.
Why are Trigonometry Formulas Important?
Trigonometry formulas play a vital role for students in various ways:
- Quick revision for board exams and competitive exams like JEE, NEET.
- Solve for unknown sides and angles in triangles.
- Applications in engineering, architecture, navigation, and daily life situations.
- Form the basis for advanced maths and physics concepts.
Basic Trigonometric Ratios
Here’s a helpful table to understand the six basic trigonometric ratios for a right-angled triangle, where θ is one of the non-right angles:
Trigonometric Ratios Table
| Ratio | Formula | Reciprocal Pair |
|---|---|---|
| sin θ | Opposite / Hypotenuse | cosec θ |
| cos θ | Adjacent / Hypotenuse | sec θ |
| tan θ | Opposite / Adjacent | cot θ |
| cosec θ | Hypotenuse / Opposite | sin θ |
| sec θ | Hypotenuse / Adjacent | cos θ |
| cot θ | Adjacent / Opposite | tan θ |
These basic ratios are the foundation of most trigonometry formulas.
List of Important Trigonometry Formulas
Below is the Trigonometry Formulas List that covers basic identities, reciprocal, ratio formulas, sum/difference, and more, essential for exams and quick reference.
Fundamental Trigonometric Identities
| Type | Formula |
|---|---|
| Pythagorean Identities | sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = cosec²θ |
| Reciprocal Identities | sinθ = 1/cosecθ cosθ = 1/secθ tanθ = 1/cotθ cosecθ = 1/sinθ secθ = 1/cosθ cotθ = 1/tanθ |
| Negative Angle Identities | sin(-θ) = -sinθ cos(-θ) = cosθ tan(-θ) = -tanθ cot(-θ) = -cotθ sec(-θ) = secθ cosec(-θ) = -cosecθ |
| Complementary Angle | sin(90°−θ) = cosθ cos(90°−θ) = sinθ tan(90°−θ) = cotθ cot(90°−θ) = tanθ sec(90°−θ) = cosecθ cosec(90°−θ) = secθ |
| Sum & Difference |
sin(A±B) = sinAcosB ± cosA sinB cos(A±B) = cosAcosB ∓ sinA sinB tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB) |
| Double Angle |
sin2θ = 2sinθcosθ cos2θ = cos²θ – sin²θ = 2cos²θ−1 = 1−2sin²θ tan2θ = 2tanθ/(1−tan²θ) |
| Triple Angle |
sin3θ = 3sinθ − 4sin³θ cos3θ = 4cos³θ − 3cosθ |
| Product to Sum |
sinA sinB = ½ [cos(A–B)–cos(A+B)] cosA cosB = ½ [cos(A–B)+cos(A+B)] sinA cosB = ½ [sin(A+B)+sin(A–B)] |
This list is commonly used for board exams, JEE, NEET and higher-level problems.
Class-wise Trigonometry Formulas
Class 10
For class 10, focus on right triangle ratio formulas and Pythagorean identities. You can find a detailed guide and worked questions at Trigonometry for Class 10.
Class 11
Class 11 includes basic identities, compound angle formulas, double angle, and transformation formulas. Practice with theory and questions at Trigonometric Functions.
JEE/NEET Trigonometry Formulas
For JEE/NEET, advanced concepts like multiple angle, transformation, and inverse trigonometric identities are crucial. Refer to topic-specific notes at JEE Main Maths - Trigonometry and JEE Advanced Maths - Trigonometry Revision Notes for fast revision.
Trigonometric Table for Standard Angles
Here’s a quick lookup table for common angles, widely asked in exams:
| Degrees | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | 1/√2 | 1/√2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | ∞ |
For more angles, see the complete Trigonometry Table on Vedantu.
Applications of Trigonometry Formulas
Trigonometry formulas are used in solving heights and distances, navigation, physics, engineering, astronomy, and even music. They help find unknown lengths, analyze waves, and calculate forces. Explore real-world examples at Application of Trigonometry.
Worked Example – Solving a Problem
Let’s solve a classic exam question:
2. Using the Pythagorean identity: sin²θ + cos²θ = 1
→ (1/2)² + cos²θ = 1
→ 1/4 + cos²θ = 1
→ cos²θ = 1 – 1/4 = 3/4
→ cos θ = √(3/4) = √3/2 (since θ is acute, cos θ is positive)
3. tan θ = sin θ / cos θ = (1/2) / (√3/2) = 1/√3
Final Answers: cos θ = √3/2, tan θ = 1/√3
Practice Problems
- Prove that tan²θ + 1 = sec²θ using identities.
- If sinA = 3/5 and A is an acute angle, find cosA and tanA.
- Simplify: sin(60° + 30°).
- If secθ = 5/3, find all other trigonometric ratios of θ.
Common Mistakes to Avoid
- Confusing the reciprocal pairs (e.g., sec vs. cos, cosec vs. sin).
- Using degree and radian values interchangeably without conversion.
- Applying identities incorrectly (e.g., mixing up sum and difference formulas).
- Ignoring signs in different quadrants.
Download Trigonometry Formulas PDF
For offline study and quick revision, download the trigonometry formulas as a printable sheet or PDF. Find more formula sheets and revision guides at Vedantu’s resources. (Search for JEE/CBSE-specific PDF sheets directly on the Vedantu platform.)
Frequently Asked Questions
Q1: What are the main trigonometry formulas?
A: The main formulas are the basic trigonometric ratios, Pythagorean identities, sum/difference identities, and reciprocal relations.
Q2: How can I download a trigonometry formulas list PDF?
A: Visit Vedantu or use their resources to download class-wise and competitive exam PDFs.
Q3: Which formulas are most important for class 10 board exams?
A: Focus on basic ratios, Pythagoras theorem, and values for standard angles.
Related Concepts & More Help
To strengthen your understanding, explore these helpful topics:
- Introduction to Trigonometry
- Trigonometric Identities (Class 10)
- Trigonometric Ratios of Complementary Angles
- Trigonometric Functions
- Trigonometric Equations
- Height and Distance
- Law of Sines
- Law of Cosines
We explored the idea of Trigonometry Formulas List, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts. Regularly revisit these sheets for revision and exam success!
FAQs on Complete Trigonometry Formulas List with Examples & PDF
1. What is the list of important trigonometry formulas for CBSE students?
The trigonometry formulas list includes basic trigonometric ratios (sin, cos, tan, cosec, sec, cot), Pythagorean identities, angle sum and difference formulas, and reciprocal identities. These formulas form the foundation for solving triangle problems in CBSE syllabus.
2. Why are trigonometry formulas important for board and competitive exams?
Trigonometry formulas are vital because they help in quick calculation of angles and sides in triangles, which is essential for board exams and competitive tests like JEE and NEET. They also support solving geometry problems, height and distance, and real-life applications efficiently.
3. How do basic trigonometric ratios differ from trigonometric identities?
Basic trigonometric ratios are the primary functions based on triangle sides: sin, cos, tan, cosec, sec, and cot. Trigonometric identities are equations involving these ratios, like the Pythagorean identity (sin²θ + cos²θ = 1) and angle sum/difference formulas, used to simplify and prove expressions.
4. Are the trigonometry formulas the same for Class 10, Class 11, and JEE students?
While core trigonometry formulas overlap across Class 10, Class 11, and JEE, higher classes and competitive exams include more advanced formulas like multiple-angle identities, transformations, and shortcut techniques for faster problem-solving.
5. What are the common mistakes students make while learning trigonometric identities?
Students often confuse formulas due to similar-looking expressions. Key mistakes include mixing up angle sum and difference formulas, forgetting sign changes for negative angles, and misapplying reciprocal identities. Careful practice and understanding of unit circle values help reduce these errors.
6. How can trigonometry formulas be applied in real-life problems?
Applications of trigonometry formulas include calculating heights and distances, designing angles in construction, navigation, engineering, and physics. They allow precise measurements where direct methods are impractical, improving spatial understanding in practical scenarios.
7. What is the difference between reciprocal identities and Pythagorean identities in trigonometry?
Reciprocal identities relate a trigonometric function to its reciprocal, e.g., sinθ = 1/cosecθ. Pythagorean identities express relationships between squares of functions, like sin²θ + cos²θ = 1 and 1 + tan²θ = sec²θ, essential for simplifying expressions and solving equations.
8. What types of trigonometric formulas are crucial for JEE exam preparation?
The JEE trigonometry formulas focus on advanced identities including multiple-angle formulas, transformations like sine and cosine of sums/differences, product-to-sum formulas, and shortcut identities that aid in faster problem solving and scoring high in exams.
9. How does understanding unit circle values enhance the use of trigonometric formulas?
Unit circle values provide exact trigonometric ratios for common angles (0°, 30°, 45°, 60°, 90°, etc.), helping students quickly recall and apply formulas without recalculating. This understanding aids in solving problems involving angle transformations and verifying identities accurately.
10. Why should students learn inverse trigonometric functions along with basic formulas?
Inverse trigonometric functions such as sin⁻¹, cos⁻¹, tan⁻¹ allow finding an angle given its ratio, crucial for problem-solving in higher mathematics and real-world scenarios. They complement basic formulas by enabling angle determination in equations and applications.
11. Why do students often confuse sine and cosine formulas, and how can this be avoided?
Confusion arises because sine and cosine functions have similar forms but differ in sign and quadrant properties. Using the unit circle model to visualize angles, practicing sign rules for different quadrants, and memorizing fundamental differences help students avoid mixing these formulas.
12. How can memorizing trigonometry identities improve problem-solving speed?
Memorizing key trigonometric identities enables quick simplification of complex expressions and faster calculation during exams. It reduces reliance on derivations, improves accuracy, and helps in spotting patterns, making problem solving more efficient and less error-prone.
13. Can trigonometry formulas be universally applied across different syllabus boards?
Most trigonometry formulas are standard and universally applicable, but the depth and emphasis may vary across boards like CBSE, ICSE, and state syllabi. However, foundational formulas remain consistent and form the basis for advanced topics in all curricula.
14. What is the importance of angle sum and difference formulas in trigonometry?
Angle sum and difference formulas are essential for expressing trigonometric functions of combined angles in terms of single angles, aiding in solving equations involving complex angles and proving identities. They are fundamental in higher-level mathematics and competitive exams.
15. How do trigonometric identities assist in mathematical proofs and derivations?
Trigonometric identities provide equivalences that simplify expressions, allowing the transformation of complex expressions into simpler forms in proofs. They are crucial in verification, equation solving, and derivation of new formulas, deepening conceptual understanding.
