Exercise 10.2 Class 10 Solution

Exercise 10.2 Class 10 Solution: A complete walkthrough to Trigonometric Identities

This article provides a comprehensive walkthrough of the solutions for Exercise 10.Understanding trigonometric identities is crucial for further studies in mathematics and related fields like physics and engineering. 2 in Class 10 mathematics textbooks, typically focusing on trigonometric identities. We will cover each problem step-by-step, explaining the underlying principles and offering helpful tips and tricks to master these important concepts. This guide aims to not only provide the solutions but also enhance your understanding of the subject matter.

Introduction to Trigonometric Identities

Before diving into the solutions, let's refresh our understanding of trigonometric identities. Practically speaking, these are equations that are true for all values of the variables involved (excluding values that make the expressions undefined, like division by zero). They are fundamental tools for simplifying trigonometric expressions and solving trigonometric equations. Some of the most common identities you'll encounter in Exercise 10.

• sin²θ + cos²θ = 1: This is the Pythagorean identity, the cornerstone of many trigonometric manipulations.
• 1 + tan²θ = sec²θ: Derived from the Pythagorean identity by dividing by cos²θ.
• 1 + cot²θ = cosec²θ: Derived from the Pythagorean identity by dividing by sin²θ.
• sin(A + B) = sinA cosB + cosA sinB: The sine addition formula.
• cos(A + B) = cosA cosB – sinA sinB: The cosine addition formula.
• tan(A + B) = (tanA + tanB) / (1 – tanA tanB): The tangent addition formula.
• sin 2θ = 2 sinθ cosθ: The double angle formula for sine.
• cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ: The double angle formula for cosine (with three variations).
• tan 2θ = 2tanθ / (1 – tan²θ): The double angle formula for tangent.

These identities, along with algebraic manipulation, are the key tools you'll need to solve the problems in Exercise 10.2.

Step-by-Step Solutions for Exercise 10.2

The exact problems in Exercise 10.2 vary depending on the specific textbook used. Even so, the general approach and types of problems are consistent. We will illustrate the solution methods with several example problems, categorized for clarity.

Category 1: Proving Trigonometric Identities

This category involves proving that a given trigonometric expression is equal to another. This usually requires applying one or more trigonometric identities and algebraic manipulation to transform one side of the equation into the other.

Example Problem 1: Prove that sin⁴θ – cos⁴θ = sin²θ – cos²θ

Solution:

1. Factor the left-hand side: We can factor the left-hand side as a difference of squares: sin⁴θ – cos⁴θ = (sin²θ + cos²θ)(sin²θ – cos²θ)

2. Apply the Pythagorean Identity: We know that sin²θ + cos²θ = 1. Substituting this into the equation: (1)(sin²θ – cos²θ) = sin²θ – cos²θ

3. Conclusion: We have successfully transformed the left-hand side into the right-hand side, thus proving the identity.

Example Problem 2: Prove that (1 + tan A)/(1 + cot A) = tan A

Solution:

1. Express in terms of sine and cosine: Rewrite the expression using the definitions tan A = sin A / cos A and cot A = cos A / sin A:

   [(1 + sin A/cos A) / (1 + cos A/sin A)]

2. Simplify the fractions: Find a common denominator for the numerator and denominator separately:

   [ (cos A + sin A)/cos A ] / [ (sin A + cos A)/sin A]

3. Simplify further: Invert and multiply:

   [(cos A + sin A)/cos A] * [sin A/(sin A + cos A)]

4. Cancel common terms: (cos A + sin A) cancels out, leaving:

   sin A / cos A = tan A

5. Conclusion: We have shown that the left-hand side simplifies to the right-hand side.

Category 2: Solving Trigonometric Equations

This category involves finding the values of the angle(s) that satisfy a given trigonometric equation.

Example Problem 3: Solve for θ: 2sin²θ – 1 = 0

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Solution:

1. Isolate sin²θ: Add 1 to both sides and divide by 2: sin²θ = 1/2

2. Take the square root: sinθ = ±√(1/2) = ±1/√2 = ±√2/2

3. Find the angles: We know that sinθ = √2/2 when θ = 45° (π/4 radians) and sinθ = -√2/2 when θ = 225° (5π/4 radians) and their coterminal angles.

4. General Solution: The general solution considering all possible angles would be θ = 45° + 360°n, θ = 135° + 360°n, θ = 225° + 360°n, and θ = 315° + 360°n, where 'n' is any integer.

Example Problem 4: Solve cos²θ - sin²θ = 1/2

Solution:

1. Use the double-angle formula: Recall that cos 2θ = cos²θ - sin²θ. Substitute this into the equation: cos 2θ = 1/2

2. Find the angles: We know that cos 2θ = 1/2 when 2θ = 60° or 2θ = 300° (and their coterminal angles).

3. Solve for θ: Divide by 2 to find the values of θ: θ = 30° or θ = 150° (and their coterminal angles).

Category 3: Simplifying Trigonometric Expressions

This involves using identities to reduce a complex trigonometric expression to a simpler form.

Example Problem 5: Simplify: (1 – cos²A)/(1 + cos A)

Solution:

1. Use the Pythagorean Identity: Replace 1 – cos²A with sin²A (since sin²A + cos²A = 1): sin²A / (1 + cos A)

2. Factor the numerator: sin²A = sin A * sin A

3. No further simplification is immediately apparent. The expression remains in its simplified form unless further context is provided within the problem.

Explanation of Underlying Mathematical Principles

The solutions presented above heavily rely on several key mathematical principles:

• Algebraic Manipulation: Proficiency in algebraic operations such as factoring, expanding, and simplifying expressions is essential. The ability to rearrange equations and isolate variables is crucial for solving trigonometric equations.
• Trigonometric Identities: A thorough understanding of the fundamental trigonometric identities and their derivations is the foundation for solving problems in Exercise 10.2. The ability to recognize which identity to apply in a given situation is a key skill to develop.
• Unit Circle: Familiarity with the unit circle and the values of trigonometric functions for various angles (especially special angles like 0°, 30°, 45°, 60°, 90°) is crucial for solving trigonometric equations.
• General Solutions: Understanding how to express the general solution for trigonometric equations, accounting for all possible angles (including coterminal angles), is essential.

Frequently Asked Questions (FAQ)

Q: What if I get stuck on a problem?

A: Try working backward from the desired result. That said, alternatively, rewrite everything in terms of sine and cosine; this often helps simplify expressions. Remember to check your work carefully at each step.

Q: Are there any online resources that can help?

A: While we cannot provide external links here, a general search for "Trigonometric identities" or "Class 10 trigonometry solutions" may yield helpful online resources and video tutorials. Remember to cross-check information from multiple sources.

Q: How can I improve my understanding of trigonometric identities?

A: Consistent practice is key. Work through numerous problems, focusing on understanding the underlying principles rather than just memorizing solutions. Try deriving the identities yourself to better understand their relationships.

Conclusion

Mastering Exercise 10.In practice, remember, practice is crucial for developing your proficiency in trigonometry. Think about it: by consistently applying the techniques and understanding the underlying principles, you will build a strong foundation for more advanced mathematical concepts. The step-by-step solutions provided in this article offer a practical guide to tackling various problem types. 2 requires a solid understanding of trigonometric identities and algebraic manipulation. In real terms, don't hesitate to revisit the fundamental identities and practice regularly to ensure you solidify your understanding. With dedication and effort, you will excel in this important area of mathematics.