NCERT Solutions Class 10 Maths Chapter 8 Introduction to Trigonometry
The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). In fact, trigonometry is the study of relationships between the sides and angles of a triangle. The earliest known work on trigonometry was recorded in Egypt and Babylon. Early astronomers used it to find out the distances of the stars and planets from the Earth. Even today, most of the technologically advanced methods used in Engineering and Physical Sciences are based on trigonometric concepts
The Introduction to Trigonometry Class 10th chapter introduces the basic trigonometric ratios-sine, cosine, and tangent-based on right-angled triangles. Through NCERT Class 10 Maths Solutions, students will easily find the step-by-step solution for all problems present in the NCERT textbook so that they can study how to solve those problems accurately. Class 10 Maths Chapter 8 Solutions are designed to strengthen students' conceptual understanding and problem-solving skills, ensuring better preparation for exams.
1.0NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry : Free PDF
Students can download NCERT Solutions pdf for class 10 Maths Chapter 8 from the link below:
NCERT Solutions for Class 10 Maths Chapter 8 - Introduction to Trigonometry
2.0NCERT Solutions Class 10 Maths Chapter 8 Introduction to Trigonometry: All Exercises
Solving NCERT Textbook exercises can help students grasp basic understanding of the key concepts as it provides a wide range of problems covering all the topics.
NCERT Solutions Class 10 Maths Chapter 8: Exercises
3.0What Will Students Learn in Chapter 8: Introduction to Geometry?
In Maths Class 10 Introduction To Trigonometry chapter, you will study some ratios of the sides of a right triangle with respect to its acute angles, called trigonometric ratios of the angle.
Students will also learn about the relationships between these ratios and how to apply them to solve various problems involving angles and sides of triangles
Discussions will be restricted to acute angles only. However, these ratios can be extended to other angles also.
You will also define the trigonometric ratios for angles of measure 0° and 90°.
Calculate trigonometric ratios for some specific angles and establish some identities involving these ratios, called trigonometric identities.
4.0Important Topics of Class 10 Math Chapter 8 Introduction of Trigonometry
Chapter 8, Introduction to Trigonometry, introduces students to the fundamental trigonometric ratios in a right-angled triangle: sine (sin), cosine (cos), and tangent (tan), as well as their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). The chapter also covers trigonometric values for specific angles and important identities like sin2θ+cos2θ=1. These concepts provide a foundation for solving real-life problems involving angles and distances.
Trigonometric Ratios
Reciprocal Trigonometric Ratios
Trigonometric Identities
Quotient Relations
Power of Trigonometric Ratios
Trigonometric Values for Specific Angles
5.0General Outline for Class 10 Maths Chapter 8: Introduction to Trigonometry
Understanding Key Concepts
Trigonometric Ratios: These ratios relate the angles of a right-angled triangle to the lengths of its sides. For a right triangle with an angle θ:
Sine (sinθ) = HypotenuseOpposite
Cosine (cosθ)=HypotenuseAdjacent
Tangent (tanθ)=AdjacentOpposite
Reciprocal ratios:
Cosecant (cscθ)=sinθ1
Secant (secθ)=cosθ1
Cotangent (cotθ)=tanθ1
Trigonometric Ratios of Specific Angles
Standard Angles:
sin30∘=21,cos30∘=23,tan30∘=31
sin45∘=22,cos45∘=22,tan45∘=1
sin60∘=23,cos60∘=21,tan60∘=3
sin90∘=1,cos90∘=0,tan90∘=undefined
Important Theorems and Formulas
Pythagoras Theorem: sin2θ+cos2θ=1
Basic Trigonometric Identity:
tanθ=cosθsinθ
sec2θ=1+tan2θ
csc2θ=1+cot2θ
Trigonometric Ratios in Different Quadrants
In the first quadrant (0° to 90°), all trigonometric ratios are positive.
In the second quadrant (90° to 180°), sine and cosecant are positive.
In the third quadrant (180° to 270°), tangent and cotangent are positive.
In the fourth quadrant (270° to 360°), cosine and secant are positive.
Steps for Solving Trigonometry Problems
Step 1: Identify the given information (angle, sides of the triangle, etc.).
Step 2: Label the sides of the triangle correctly relative to the given angle.
Opposite side: The side opposite to the given angle.
Adjacent side: The side next to the angle, excluding the hypotenuse.
Hypotenuse: The longest side opposite the right angle.
Step 3: Choose the appropriate trigonometric ratio (sine, cosine, tangent, etc.) based on the sides involved in the problem.
Step 4: Apply the known values and solve the equation.
Step 5: If the problem asks for a specific angle, use the inverse trigonometric functions(sin−1,cos−1,tan−1) to find the angle.
Step 6: Verify the solution (check if the side lengths satisfy the triangle’s properties or trigonometric identities).
Common Problem Types
Finding Unknown Sides: Given one side and an angle, use the appropriate trigonometric ratio to find unknown sides of a right-angled triangle.
Finding Angles: Given two sides, use the inverse trigonometric functions to find the angles of a triangle.
Using Trigonometric Identities: Simplify expressions involving trigonometric functions or prove identities.
Applications: Real-world problems such as finding the height of an object (e.g., building or tree) using angles of elevation or depression.
6.0Benefits of Class 10 Maths Chapter 8 Introduction to Trigonometry
The Introduction to Trigonometry chapter in Class 10 Maths offers several key benefits:
Foundation for Advanced Trigonometry: It introduces basic trigonometric ratios (sine, cosine, tangent) that are crucial for higher-level studies in mathematics and physics.
Real-Life Applications: Trigonometry is widely used in fields like engineering, architecture, navigation, and astronomy to calculate distances, angles, and heights.
Problem-Solving Skills: Students develop skills to solve practical problems involving right-angled triangles.
Analytical Thinking: The chapter enhances logical thinking and the ability to apply mathematical concepts to real-world situations.
Link to Further Topics: It forms the basis for more advanced trigonometric concepts that students will encounter in higher classes.
Trigonometry is a mathematical discipline focused on the exploration of relationships between angles and sides in a right triangle. Trigonometry is necessary for solving problems involving distances and angles in right triangles, and is useful for applications in a wide variety of fields
In the Class 10 CBSE Maths exam, terms for trigonometry, specifically, the two, each typically contributes around twelve to fourteen marks, including both direct questions and applications of trigonometry. The exact weightage may vary every year.
The hardest questions in Class 10 Trigonometry are, usually, complex word problems, proofs of trigonometric identities, and questions that require multiple steps with more than one trigonometric ratio. Example: A ladder 10 meters long is placed against a wall. If the foot of the ladder is 6 meters away from the wall, find the height at which the ladder touches the wall. Use trigonometric ratios to solve the problem.