Trigonometric Identities Formula List for Class 10 Students
The concept of Trigonometric Identities Class 10 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Trigonometric Identities Class 10?
Trigonometric identities class 10 are special mathematical equations that show consistent relationships between the trigonometric ratios: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). These identities are always true for defined values of an angle and help simplify complex trigonometric expressions. You’ll find this concept applied in areas such as trigonometric ratios, problem-solving in trigonometry, and applications of trigonometry in geometry and real life.
Key Formula for Trigonometric Identities Class 10
Here are the three core trigonometric identities that you need to know for class 10:
| Identity | Formula | True For |
|---|---|---|
| Pythagorean (Sin & Cos) | sin2θ + cos2θ = 1 | 0° ≤ θ ≤ 90° |
| Tan & Sec | 1 + tan2θ = sec2θ | 0° ≤ θ < 90° |
| Cot & Cosec | 1 + cot2θ = cosec2θ | 0° < θ ≤ 90° |
Other important formulas related to trigonometric identities are also used to express one trigonometric ratio in terms of others (for example, tan θ = sin θ / cos θ and sec θ = 1 / cos θ).
Cross-Disciplinary Usage
Trigonometric identities class 10 are not only useful in Maths but also play an important role in Physics, engineering, and daily logical reasoning. Students preparing for competitive exams like JEE, NEET, and Olympiads see these identities used in questions related to waves, oscillations, and angle measurements. Vedantu lessons often extend these connections for deeper understanding.
Step-by-Step Illustration
Let’s prove that sin2A + cos2A = 1 for any angle A (0° ≤ A ≤ 90°):
1. Consider a right-angled triangle with sides: hypotenuse = AB, base = AC, perpendicular = BC.2. By Pythagoras’ theorem: AB2 = AC2 + BC2
3. Divide both sides by AB2:
4. (AB2)/(AB2) = (AC2)/(AB2) + (BC2)/(AB2)
5. 1 = (AC/AB)2 + (BC/AB)2
6. From trigonometric definitions: cos A = AC/AB and sin A = BC/AB
7. Substituting: 1 = (cos A)2 + (sin A)2
8. Or, sin2A + cos2A = 1
This identity helps to simplify many board exam problems quickly.
Speed Trick or Vedic Shortcut
Here’s a quick trick: If a question asks you to find 1 − sin2A, just recall the identity sin2A + cos2A = 1 so that:
1. 1 − sin2A = cos2A
This shortcut makes trigonometric simplifications in MCQ questions speedy and reduces calculation mistakes, especially during time-pressured board exams.
Try These Yourself
- Simplify the expression: sin2θ + cos2θ + tan2θ − sec2θ.
- If cos A = 3/5, find sin A using an identity.
- Prove that sec2P − tan2P = 1.
- Express cot θ in terms of cosec θ.
Frequent Errors and Misunderstandings
- Forgetting the squares in sin2θ or cos2θ, leading to wrong simplifications.
- Mixing up identities: e.g., using tan2θ + cos2θ = 1 (which is NOT correct).
- Missing domain restrictions (e.g., tan 90° is undefined).
Relation to Other Concepts
The idea of trigonometric identities class 10 connects closely with trigonometric ratios, complementary angles, and applications such as height and distance problems. Mastering these helps in understanding the use of formulas and angle transformations in both boards and higher classes.
Classroom Tip
A quick way to remember the core identities is the “Magic 1” trick: Every identity has 1 on one side (sin2θ + cos2θ = 1, 1 + tan2θ = sec2θ, 1 + cot2θ = cosec2θ). Vedantu teachers often draw triangle diagrams and identity charts to help students visualize and memorize quickly.
We explored trigonometric identities class 10—from definition, formula, stepwise sample, common doubts, and links to other maths topics. Continue practicing with Vedantu to become confident in solving trigonometry questions and excel in your exams.
FAQs on Trigonometric Identities for Class 10: Complete Guide
1. What are trigonometric identities in Class 10 maths?
Trigonometric identities in Class 10 are equations that are true for all values of the angle involved. They show relationships between different trigonometric ratios like sin, cos, tan, cosec, sec, and cot. These identities are fundamental to simplifying complex trigonometric expressions and solving problems.
2. How many trigonometric identities are there in Class 10?
While there are many trigonometric identities, Class 10 typically focuses on a core set of three fundamental identities and their variations: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ. Understanding these allows you to derive and apply numerous others.
3. What is the best way to memorize trigonometric identities?
The most effective way is to understand their derivation. However, for quick recall, use mnemonics, create flashcards, write them out repeatedly, or use visual aids like charts. Practicing numerous problems that apply these identities also helps solidify your memory.
4. Are these identities used in board exams and MCQs?
Absolutely! Trigonometric identities are crucial for both MCQ and longer-answer questions in board exams. They are used to simplify expressions, solve equations, and prove other trigonometric relationships. Mastering them significantly improves your exam performance.
5. Is Class 10 trigonometry difficult for average students?
With consistent practice and a structured approach, Class 10 trigonometry is manageable for most students. Focus on understanding the core concepts, mastering the identities, and solving a variety of problems. Seek help from teachers or online resources when needed.
6. How do I solve trigonometric identities?
Solving trigonometric identities involves using algebraic manipulation and the fundamental identities to simplify an expression to a known identity or a simpler form. This often involves using techniques like substitution, factoring, and applying known trigonometric ratios and properties.
7. What are some common mistakes to avoid when solving trigonometric identities?
Common mistakes include incorrect application of identities, algebraic errors, and forgetting to consider the domain and range of trigonometric functions. Carefully review your steps and check for errors in your calculations to prevent these mistakes.
8. How are trigonometric identities used in real-world applications?
Trigonometric identities have numerous real-world applications, particularly in fields like physics and engineering. They are used to model and analyze wave motion, oscillations, and other periodic phenomena, helping in the design and analysis of systems with repetitive patterns.
9. What resources can help me further practice trigonometric identities?
Numerous resources are available, including textbooks, online tutorials, and practice websites. Vedantu provides detailed notes, solved examples, and practice problems specifically designed for Class 10 students preparing for board exams. Utilize these resources to enhance your skills and understanding.
10. How do I choose the right trigonometric identity to use in a problem?
The choice depends on the specific problem. Look for terms in the expression that suggest a particular identity. For example, if you see sin²θ and cos²θ, the identity sin²θ + cos²θ = 1 is likely relevant. Practice will help you recognize patterns and choose the most efficient approach.
11. Can I use a calculator to verify trigonometric identities?
While a calculator can help check numerical values, it doesn't prove the identity for all angles. Verification requires showing the algebraic equivalence using the fundamental identities. Calculators are useful tools to check your work, but not as a replacement for proper proof.
12. What's the difference between proving and verifying a trigonometric identity?
Proving involves demonstrating the identity's truth using logical steps and established identities. Verifying typically involves substituting specific values of θ to check if the equation holds true. While verification provides support, a complete proof is necessary for rigorous mathematical demonstration.
