Math Exercise 6.3 Class 10

Mastering Math Exercise 6.3 Class 10: A Comprehensive Guide

This article provides a thorough walkthrough of Math Exercise 6.3 for Class 10 students. We'll cover the key concepts, provide step-by-step solutions to example problems, and address common student queries. This exercise typically focuses on trigonometric identities and their applications in solving equations and simplifying expressions. Mastering this section is crucial for building a strong foundation in trigonometry and succeeding in higher-level mathematics. We'll break down the complexities, ensuring you understand not just the how but also the why behind each step.

Introduction to Trigonometric Identities

Before diving into Exercise 6.3, let's refresh our understanding of trigonometric identities. These are equations involving trigonometric functions (like sin, cos, tan, etc.) that are true for all values of the involved angles. These identities are fundamental tools for simplifying complex trigonometric expressions and solving trigonometric equations. Some of the most important identities include:

• Basic Identities:
  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ
  • sec θ = 1 / cos θ
  • cosec θ = 1 / sin θ
• Pythagorean Identities:
  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = cosec²θ
• Sum and Difference Identities:
  • sin(A + B) = sinA cosB + cosA sinB
  • sin(A - B) = sinA cosB - cosA sinB
  • cos(A + B) = cosA cosB - sinA sinB
  • cos(A - B) = cosA cosB + sinA sinB
• Double Angle Identities: These are derived from the sum identities by setting A = B.
  • sin2θ = 2sinθcosθ
  • cos2θ = cos²θ - sin²θ = 1 - 2sin²θ = 2cos²θ - 1

Understanding and remembering these identities is crucial for successfully completing Exercise 6.3. We will see how they are applied in the problems below.

Step-by-Step Solutions to Example Problems from Exercise 6.3

Let's tackle some typical problems found in Exercise 6.3, demonstrating the application of the identities mentioned above. Remember, the specific problems will vary depending on your textbook, but the underlying principles remain the same.

Example Problem 1: Prove the identity sin⁴θ - cos⁴θ = sin²θ - cos²θ

Solution:

We can factor the left-hand side using the difference of squares:

sin⁴θ - cos⁴θ = (sin²θ + cos²θ)(sin²θ - cos²θ)

Since sin²θ + cos²θ = 1 (Pythagorean identity), the equation simplifies to:

(1)(sin²θ - cos²θ) = sin²θ - cos²θ

This matches the right-hand side, thus proving the identity.

Example Problem 2: Prove the identity (1 + tan²A) / (1 + cot²A) = tan²A

Solution:

We use the Pythagorean identities: 1 + tan²A = sec²A and 1 + cot²A = cosec²A. Substituting these, we get:

(sec²A) / (cosec²A) = (1/cos²A) / (1/sin²A) = sin²A / cos²A = tan²A

This proves the given identity.

Example Problem 3: Solve the equation 2cos²θ - 3cosθ + 1 = 0 for 0 ≤ θ ≤ 2π

Solution:

This is a quadratic equation in terms of cosθ. We can factor it as:

(2cosθ - 1)(cosθ - 1) = 0

This gives two possible solutions:

• 2cosθ - 1 = 0 => cosθ = 1/2 => θ = π/3, 5π/3
• cosθ - 1 = 0 => cosθ = 1 => θ = 0, 2π

Therefore, the solutions for θ in the given range are 0, π/3, 5π/3, and 2π.

Example Problem 4: Simplify the expression (sinA + cosA)² + (sinA - cosA)²

Solution:

Expanding the squares, we get:

(sin²A + 2sinAcosA + cos²A) + (sin²A - 2sinAcosA + cos²A)

Combining like terms, we have:

2sin²A + 2cos²A = 2(sin²A + cos²A) = 2(1) = 2

These examples illustrate the various techniques used to solve problems in Exercise 6.3. The key is to carefully select the appropriate trigonometric identities and apply algebraic manipulation to simplify expressions or solve equations.

Addressing Common Student Challenges in Exercise 6.3

Many students find Exercise 6.3 challenging due to:

• Memorization of Identities: Remembering all the identities can be overwhelming. Focus on understanding the relationships between the functions and deriving identities when needed, rather than rote memorization. Practice is key here.
• Algebraic Manipulation: Solving trigonometric equations often requires skillful algebraic manipulation. Practice simplifying algebraic expressions and solving quadratic equations.
• Selecting the Right Identity: Knowing which identity to apply in a given situation is crucial. Look for patterns and relationships between the terms in the given expression or equation.

To overcome these challenges:

• Consistent Practice: Regular practice is the best way to improve your understanding and skills. Work through as many problems as possible.
• Seek Clarification: Don't hesitate to ask your teacher or tutor if you're stuck on a particular problem.
• Break Down Complex Problems: Divide complex problems into smaller, manageable steps. Focus on one step at a time.
• Utilize Online Resources: While you shouldn't directly copy answers, reputable educational websites and videos can provide further explanation and practice problems.

Frequently Asked Questions (FAQ)

Q: What if I don't remember all the trigonometric identities?

A: It's not essential to memorize every single identity. Focus on the most fundamental ones (Pythagorean identities, basic identities, and perhaps the sum/difference identities for sine and cosine). You can often derive others from these core identities.

Q: How do I know which identity to use?

A: Look for patterns in the given expression or equation. If you see sin²θ + cos²θ, you know you can substitute 1. If you see a term like sin2θ, consider using the double angle identity. Practice will help you recognize these patterns more easily.

Q: What should I do if I get stuck on a problem?

A: Try a different approach. If one method isn't working, try manipulating the expression in a different way or using a different identity. If you're still stuck, seek help from your teacher or a classmate.

Q: Is it important to show all my work?

A: Absolutely! Showing your steps is crucial, not only for getting the correct answer but also for demonstrating your understanding of the concepts involved. It allows you to identify where you might have made a mistake.

Conclusion

Mastering Math Exercise 6.3 requires a strong understanding of trigonometric identities and algebraic manipulation. By consistently practicing, understanding the underlying concepts, and seeking help when needed, you can confidently tackle these problems and build a solid foundation in trigonometry. Remember, practice makes perfect! The more you work through these problems, the more comfortable and proficient you will become. Don't get discouraged if you struggle at first; perseverance and a focused approach will lead to success.