Maths Ex 6.3 Class 10

Mastering Math: A Deep Dive into Class 10 Ex 6.3 (Trigonometry)

This comprehensive guide delves into the intricacies of Class 10 Exercise 6.3, focusing on the application of trigonometric ratios and identities. We'll break down the concepts, provide step-by-step solutions to example problems, and address frequently asked questions. This exercise typically covers trigonometric ratios of complementary angles, and proving trigonometric identities. Understanding this chapter is crucial for building a strong foundation in trigonometry and succeeding in higher-level mathematics. By the end of this article, you'll not only be able to solve the problems in Ex 6.3 but also grasp the underlying principles of trigonometry.

Introduction to Trigonometry and Exercise 6.3

Trigonometry, derived from the Greek words "trigonon" (triangle) and "metron" (measure), is the branch of mathematics that deals with the relationships between angles and sides of triangles. Exercise 6.3 typically builds upon the foundational concepts of trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) and introduces the application of these ratios to solving problems involving right-angled triangles and proving trigonometric identities. It emphasizes the relationships between trigonometric ratios of complementary angles (angles that add up to 90°). This exercise is a crucial stepping stone towards understanding more complex trigonometric concepts in higher classes.

Understanding Trigonometric Ratios

Before we tackle the problems in Ex 6.3, let's refresh our understanding of the six basic trigonometric ratios:

• Sine (sin): sin θ = Opposite side / Hypotenuse
• Cosine (cos): cos θ = Adjacent side / Hypotenuse
• Tangent (tan): tan θ = Opposite side / Adjacent side
• Cosecant (cosec): cosec θ = 1 / sin θ = Hypotenuse / Opposite side
• Secant (sec): sec θ = 1 / cos θ = Hypotenuse / Adjacent side
• Cotangent (cot): cot θ = 1 / tan θ = Adjacent side / Opposite side

These ratios are defined with respect to an acute angle θ in a right-angled triangle. Remember the mnemonic "SOH CAH TOA" to help you remember the ratios: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

Trigonometric Ratios of Complementary Angles

A key concept explored in Ex 6.3 is the relationship between trigonometric ratios of complementary angles. Complementary angles are two angles whose sum is 90°. The following relationships hold true:

• 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

Understanding these identities is vital for solving many problems in this exercise.

Step-by-Step Solutions to Example Problems (Ex 6.3)

Let's work through some example problems typical of those found in Class 10 Ex 6.3. The specific problems will vary depending on the textbook used, but the underlying principles remain consistent.

Example 1: Proving a Trigonometric Identity

Prove that: sin² θ + cos² θ = 1

Solution:

This is a fundamental trigonometric identity. We can prove it using the Pythagorean theorem. Consider a right-angled triangle with hypotenuse 'h', opposite side 'o', and adjacent side 'a'. According to the Pythagorean theorem: h² = o² + a²

Dividing by h²: 1 = (o²/h²) + (a²/h²)

Substituting the trigonometric ratios: 1 = sin² θ + cos² θ

Therefore, the identity is proven.

Example 2: Using Trigonometric Ratios of Complementary Angles

If sin 3A = cos (A - 26°), where 3A is an acute angle, find the value of A.

Solution:

We know that sin θ = cos (90° - θ). Therefore, we can write:

sin 3A = cos (90° - 3A)

Given sin 3A = cos (A - 26°), we can equate the two expressions for sin 3A:

cos (90° - 3A) = cos (A - 26°)

Since the cosine function is equal for two angles if the angles are equal or differ by a multiple of 360°, we have:

90° - 3A = A - 26° or 90° - 3A = 360°n + (A -26°) where n is an integer

Solving the first equation:

90° + 26° = 4A

116° = 4A

A = 29°

Solving the second equation:

90° +26° = 4A +360n°

116° = 4A +360n°

For n=0 we have A = 29° which is a valid solution since 3A =87° is acute

For n=-1, 4A = 476, A = 119 which gives 3A = 357°, not acute

Hence, only A=29° is a valid solution.

Example 3: Solving a Problem Involving Trigonometric Ratios

In a right-angled triangle ABC, right-angled at B, if tan A = 1/√3, find the value of: sin A cos C + cos A sin C.

Solution:

If tan A = 1/√3, then A = 30° (since tan 30° = 1/√3). In a right-angled triangle, A + C = 90°. Therefore, C = 90° - A = 90° - 30° = 60°.

Now substitute the values:

sin A cos C + cos A sin C = sin 30° cos 60° + cos 30° sin 60°

= (1/2)(1/2) + (√3/2)(√3/2)

= 1/4 + 3/4 = 1

This problem showcases the application of trigonometric ratios in solving practical problems related to right-angled triangles.

Explanation of Important Concepts

1. Trigonometric Identities: These are equations that are true for all values of the variable (angle). The identity sin² θ + cos² θ = 1 is a fundamental example. Understanding and proving these identities is critical for more advanced trigonometric work.

2. Complementary Angles: Mastering the relationships between trigonometric ratios of complementary angles is essential for simplifying expressions and solving equations. This concept allows us to express a trigonometric function in terms of its cofunction.

3. Solving Trigonometric Equations: Exercise 6.3 often involves solving equations where the unknown is an angle. This usually requires applying trigonometric identities and the properties of complementary angles to isolate the unknown and find its value.

Frequently Asked Questions (FAQ)

Q: Why is it important to learn Exercise 6.3?

A: Exercise 6.3 forms the foundation for understanding more advanced trigonometric concepts. The skills learned in this exercise, such as using trigonometric identities and understanding complementary angles, are crucial for solving complex problems in higher-level mathematics and other related fields like physics and engineering.

Q: What are some common mistakes students make in this exercise?

A: Common mistakes include:

• Incorrectly applying trigonometric ratios.
• Confusing the relationships between trigonometric ratios of complementary angles.
• Making algebraic errors while solving equations.
• Not understanding the context of the problem before attempting a solution.

Q: How can I improve my understanding of this exercise?

A:

• Practice: Solve a wide variety of problems from different sources, including textbooks and online resources.
• Understand the concepts: Don't just memorize formulas, understand the underlying principles.
• Seek help: If you're stuck, ask your teacher, tutor, or classmates for assistance.
• Review regularly: Consistent review is key to solidifying your understanding.

Conclusion

Mastering Class 10 Ex 6.3 requires a solid understanding of trigonometric ratios, identities, and the relationships between trigonometric functions of complementary angles. By working through example problems and practicing regularly, you can build confidence and proficiency in solving trigonometric problems. Remember to focus on understanding the underlying concepts, not just memorizing formulas. This exercise is a vital stepping stone to more advanced mathematical concepts, making it a crucial part of your mathematical journey. With dedication and practice, you can achieve mastery over this important topic and build a strong foundation for future success in mathematics.