Exercise 8.1 Class 10 Solutions

Exercise 8.1 Class 10 Solutions: A full breakdown to Introduction to Trigonometry

This article provides comprehensive solutions and explanations for Exercise 8.We'll walk through each problem, offering detailed step-by-step solutions and clarifying the underlying trigonometric concepts. 1 of Class 10 Mathematics, focusing on the introduction to trigonometry. Understanding this exercise is crucial for mastering trigonometry, a fundamental topic in higher mathematics and various scientific fields. This guide aims to not only provide answers but also to build a strong conceptual foundation.

Introduction to Trigonometry: Understanding the Basics

Trigonometry, derived from the Greek words "trigonon" (triangle) and "metron" (measure), deals with the relationships between angles and sides of triangles. In Class 10, the introduction typically focuses on right-angled triangles, where one angle is 90 degrees. Three primary trigonometric ratios are introduced:

• Sine (sin): The ratio of the side opposite the angle to the hypotenuse. sin θ = Opposite/Hypotenuse
• Cosine (cos): The ratio of the side adjacent to the angle to the hypotenuse. cos θ = Adjacent/Hypotenuse
• Tangent (tan): The ratio of the side opposite the angle to the side adjacent to the angle. tan θ = Opposite/Adjacent

Understanding these ratios is key to solving problems in Exercise 8.1 and beyond. Remember that the hypotenuse is always the longest side of a right-angled triangle, and its length is crucial for calculating the other ratios.

Exercise 8.1: Detailed Solutions and Explanations

Let's now tackle the problems in Exercise 8.In practice, the exact questions may vary slightly depending on the textbook used, but the underlying principles remain the same. We'll illustrate with example problems and provide step-by-step solutions. Day to day, 1, assuming a standard curriculum. Remember to always draw a clear diagram to visualize the problem before starting calculations.

Example Problem 1: Find the values of sin A, cos A, and tan A in the given right-angled triangle ABC, where angle B is 90 degrees, AB = 4 cm, BC = 3 cm, and AC = 5 cm.

Solution:

1. Draw the Diagram: Draw a right-angled triangle ABC, labeling the sides AB, BC, and AC with their respective lengths (4 cm, 3 cm, and 5 cm). Angle B is the right angle (90 degrees).

2. Identify Opposite, Adjacent, and Hypotenuse: With respect to angle A:

   • Opposite side (to angle A) = BC = 3 cm
   • Adjacent side (to angle A) = AB = 4 cm
   • Hypotenuse = AC = 5 cm

3. Apply Trigonometric Ratios:

   • sin A = Opposite/Hypotenuse = 3/5
   • cos A = Adjacent/Hypotenuse = 4/5
   • tan A = Opposite/Adjacent = 3/4

Example Problem 2: If sin A = ¾, find cos A and tan A.

Solution:

This problem requires understanding the Pythagorean identity: sin²A + cos²A = 1

1. Use the Pythagorean Identity: We are given sin A = ¾. So, sin²A = (¾)² = 9/16.

2. Solve for cos A:

   • 9/16 + cos²A = 1
   • cos²A = 1 - 9/16 = 7/16
   • cos A = √(7/16) = √7/4 (We take the positive square root since A is an acute angle)

3. Calculate tan A:

   • tan A = sin A / cos A = (¾) / (√7/4) = 3/√7 = 3√7/7 (Rationalizing the denominator)

Example Problem 3: Given tan A = 1, find sin A and cos A.

Solution:

1. Consider a Right-Angled Triangle: When tan A = 1, it means the opposite and adjacent sides are equal. Let's assume both are 'x' cm.

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2. Use Pythagoras Theorem: The hypotenuse (h) can be found using Pythagoras theorem: x² + x² = h² => h² = 2x² => h = x√2

3. Calculate sin A and cos A:

   • sin A = Opposite/Hypotenuse = x / (x√2) = 1/√2 = √2/2
   • cos A = Adjacent/Hypotenuse = x / (x√2) = 1/√2 = √2/2

Example Problem 4 (More Challenging): In a right-angled triangle ABC, right-angled at B, if tan A = 1/√3, find the value of: (i) sin A cos C + cos A sin C (ii) cos A cos C – sin A sin C

Solution:

This problem requires a deeper understanding of trigonometric identities and the relationship between angles in a right-angled triangle.

1. Visualize the Triangle: Draw the triangle and label the sides based on tan A = 1/√3. This indicates that the opposite side to angle A is 1 and the adjacent side is √3. The hypotenuse would be 2 (using Pythagoras theorem).

2. Deduce the Values:

   • sin A = 1/2
   • cos A = √3/2
   • Since angles A and C are complementary (A + C = 90°), sin C = cos A = √3/2 and cos C = sin A = 1/2

3. Substitute and Solve:

   • (i) sin A cos C + cos A sin C = (1/2)(1/2) + (√3/2)(√3/2) = ¼ + ¾ = 1
   • (ii) cos A cos C – sin A sin C = (√3/2)(1/2) – (1/2)(√3/2) = 0

These examples demonstrate the application of trigonometric ratios and identities in solving problems from Exercise 8.1. Remember to always check your work and ensure your answers make sense within the context of the problem.

Explanation of Scientific Principles

The trigonometric ratios are fundamental building blocks in many scientific and engineering fields. They are used extensively in:

• Physics: Calculating forces, velocities, and accelerations in various contexts, including projectile motion, wave mechanics, and optics.
• Engineering: Designing structures, calculating stresses and strains, and analyzing mechanical systems. Trigonometry is crucial in surveying and navigation.
• Computer Graphics: Creating realistic images and animations by manipulating coordinates and angles in three-dimensional space.

A strong foundation in trigonometry is essential for understanding these applications.

Frequently Asked Questions (FAQs)

Q1: What is the difference between sin, cos, and tan?

A1: Sin, cos, and tan are trigonometric ratios that relate the angles and sides of a right-angled triangle. They are defined as: * sin = Opposite/Hypotenuse * cos = Adjacent/Hypotenuse * tan = Opposite/Adjacent

Q2: How do I remember the trigonometric ratios?

A2: Many students use mnemonics like "SOH CAH TOA" (Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent) to help remember them.

Q3: What are complementary angles?

A3: Complementary angles are two angles whose sum is 90 degrees. In a right-angled triangle, the two acute angles are complementary.

Q4: What are some common mistakes to avoid when solving trigonometry problems?

A4: Common mistakes include: * Incorrectly identifying the opposite, adjacent, and hypotenuse sides. * Making calculation errors. * Using incorrect formulas or identities. * Not drawing a clear diagram to visualize the problem.

Conclusion

Exercise 8.Even so, remember to practice regularly, and don't hesitate to seek clarification on any confusing concepts. Which means by understanding the basic trigonometric ratios (sin, cos, tan) and applying them systematically, students can build a solid foundation and confidently tackle more complex problems. 1 of Class 10 Mathematics lays the groundwork for understanding trigonometry. Mastering this exercise is crucial for future mathematical studies and its applications in various scientific and engineering disciplines. With consistent effort and understanding, trigonometry will become a manageable and even enjoyable subject.