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Xam Idea Class 10 Maths Chapter 10 Solutions: Introduction To Trigonometry

Xam Idea Class 10 Maths Solutions Chapter 10 ‘Introduction to Trigonometry’ has been compiled in order to provide you with professional guidance for solving the exercise questions in this chapter of Xam Idea Solutions. The chapter will introduce you to the concepts of trigonometry and is based on the latest syllabus of CBSE for Class 10.

In this chapter, you will learn about the trigonometric ratios of angles in a triangle. You will learn the values of the trigonometric ratios for some specific angles such as 300, 450, and so on. These ratios are related to one another through some important trigonometric identities and rules for complementary angles which are also covered in this chapter.

There are 46 questions in the chapter covered in 6 exercises. Each exercise covers the topics of this chapter such as trigonometric ratios and identities related to them. You will get to solve exercises based on different marking patterns in board exams such as very short and short answer type questions, long answer questions, High Order Thinking Skills, and value-based questions. Practising these questions will assist you in getting an idea about how to manage your time well in the exam hall. The exercise questions in this chapter are apt for quick revision. It is recommended that you solve these questions thoroughly for a significant boost in your score and rank.

Important Topics for Xam Idea Class 10 Maths Solutions Chapter 10: Introduction to Trigonometry

Introduction

While looking at a building or a tree, our line of vision completes a right-angled triangle, of which the properties and ratios have been discussed and introduced in this chapter, ‘Introduction to Geometry’. These concepts will help you build a strong base so that you will be able to understand the applications of trigonometric concepts easily in the chapters ahead and in your higher classes.

Trigonometry comes originally from a combination of the Greek words ‘tri’ which means three, gon meaning sides and metron meaning measure. Therefore, trigonometry discusses the relationship between the sides and angles of a given right-angled triangle.

Trigonometric Ratios

Let us consider a right-angled triangle ABC right-angled at B such that AB is the side opposite to the angle C and BC is the side adjacent to the angle C. Now, we can define different specific trigonometric ratios with respect to this triangle ABC. These ratios are mentioned as follows:

These ratios are fixed for every triangle and are abbreviated as sin A, cos A, Tan A, Cosec A, Sec A, and Cot A.

Overall, we may conclude that the trigonometric ratios of an acute angle are used to express a relationship between the sides of the given triangle and its angles.

We should note an important point which says that the values of these trigonometric functions remain the same even if (a) the lengths of sides of the given triangle vary and (b) the angle remains the same.  

We have studied in previous classes that the angles that add up to form a sum of 900 are termed as complementary angles. We will see in the sections ahead, how these trigonometric ratios are related for complementary angles.

Trigonometric Ratios of Some Specific angles

For the triangle and trigonometric ratios described above, here are some important results to remember:

Complementary Angles and their Trigonometric Ratios

Let us consider a right-angled triangle ABC right-angled at B such that AB is the side opposite to the angle C and BC is the side adjacent to the angle C. Now, we know that ∠A+∠B=90, therefore they make a complementary pair. For complementary angles, the rules mentioned below must be followed:

 

 

 

Important Trigonometric Identities

When an equation includes trigonometric ratios, then it is termed as a trigonometric identity. Considering a right-angled triangle ABC right-angled at B such that AB is the side opposite to the angle C and BC is the side adjacent to the angle C, following identities hold true:

  • The sum of the squares of the cosine of angle A and sine of angle A is equivalent to unity. Mathematically, it can be represented as,

           cos2A+sin2A=1

  • Square of the tangent of A is one less than the square of the secant of the angle A. Mathematically, it can be denoted as,

           1+tan2A=sec2A

  • Square of cotangent of angle A is one less than the square of cosecant of angle A. The mathematical notation of the identity is as follows cot2A+1=cosec2A

These identities can be proved using the Pythagoras theorem in which the square of the hypotenuse is equal to the sum of the squares of the base and height of a given right-angled triangle.

Exercise Discussion of Xam Idea Class 10 Maths Solutions Chapter 10: Introduction to Trigonometry

Very Short Answer Type Question

This section covers a total number of 7 questions where you have to answer in only one line. The probable questions in this exercise will need you to revise the identities thoroughly besides learning the complementary trigonometric ratios.

Short Answer Type Questions

There are two sections of short answer type questions consisting of 15 and 8 questions respectively. The questions are miscellaneous type questions in which you will need to find other trigonometric ratios from a given ratio or application of the identities learnt in this chapter. You will also get to use the property of complementary angles in these exercises.

Long Answer Type Questions

This section contains 7 questions where the questions are based on diagrams of right-angled triangles and you will be required to evaluate various trigonometric ratios. You might also be required to use the trigonometric identities discussed in this chapter to evaluate different dimensions.

HOTS (High Order Thinking Skills)

In this exercise, there are 5 questions in which you will be required to prove various complex trigonometric identities. These answers to these questions will require an in-depth knowledge of the three basic identities learnt in this chapter. Using these identities, you will be able to prove further important results for future use.

Value-Based Questions

The questions in this exercise are based on practical life situations such as when the observer stands on the bank of a river, or when a tree is being observed. You will also get to solve questions based on a ladder supported by a wall and similar instances. All important situations are covered in a total of 4 questions.

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