Title 8. INTRODUCTION TO TRIGNOMETRY.pdf ✅

CLASS - X MATHEMATICS 1 FOUNDATION 8. INTRODUCTION TO TRIGONOMETRY VOL -II Application of Trigonometry Different units of measuring Angles Relation between Degree & Radian Trigonometric Ratios Trigonometric rations of complementary angles Trigonometric identities Signs of Trigonometric functions in different quadrants Trigonometry has its application in astronomy, geography, surveying, engineering and navigation etc. In the past, astronomers used it to find out the distance of stars and plants from the earth. Even now, the advanced technology used in Engineering are based on trigonometrical concepts. In this chapter, we will define trigonometric ratios of angles in terms of ratios of sides of a right triangle. We will also define trigonometric ratios of angles of 0°, 30°, 45°, 60° and 90°. We shall also establish some identities involving these ratios. SYNOPSIS – 1 1.1 TRIGONOMETRY What is Trigonometry? It is the combination of three Greek words. Tri + gonia + metron ↓ ↓ ↓ Three angle measurement Thus, Trigonometry means three-angle measurement. It is an analytical study of three- angled geometrical figures in one plane. Application of Trigonometry The study of trigonometry is of great importance in several fields. 1. In construction of machines. 2. In finding the distance between the heavenly bodies. 3. In finding the heights of mountains, towers etc. 4. In land survey. 5. In finding the depth of rivers and oceans. 6. In measuring the width of rivers. Angle Consider a ray OA. If this ray rotates about Its end point O it takes the position OB. CHAPTER 8 INTRODUCTION TO TRIGONOMETRY
2 MATHEMATICS VOL - II CLASS - X 8. INTRODUCTION TO TRIGONOMETRY FOUNDATION Then we say that AOB has been generated. The figure obtained by two rays with same initial point is called an angle. The common initial point is called the vertex of the angle and the rays forming the angle are called it’s arms or sides. The symbol used to denote an angle is ‘∠’. Thus, we name the above angle as ∠AOB or ∠BOA. However, sometimes we may name an angle by it’s vertex alone, as ∠O. Ex. In a right-angled triangle ∠ABC = 90° is an angle ∠BAC = 60° is an angle ∠BCA = 30° is an angle 1.2 DIFFERENT UNITS OF MEASURING ANGLES There are three known systems for measurement of angles. They are i) Sexagesimal system (or) English system ii) Centesimal system (or) French system iii) Circular system (or) universally accepted system. SEXAGESIMAL SYSTEM (OR) ENGLISH SYSTEM It is a system of measurement of angles in degrees. If a right angle is divided into 90 equal parts then each part is called a degree. It is written as 90° 1 right angle = 90° Further, if one degree is divided into 60 equal parts, each part is called a minute and is written as 60| 1 degree = 60 minutes = 60| Again, if one minute is further divided into 60 equal parts, each part is called a second and it is written as 60|| 1 minute = 60 seconds = 60|| degree = 60 minutes = 3600 seconds 1| = 60|| 1° = 60| = 3600|| Examples: 1. The number of minutes in 4.5 degrees 4.5° = 4.5 × 60° = 270| Therefore 4.5 degrees = 270 minutes 2. The number of seconds in 3.2 degrees, 3.2° = 3.2 × 60 = 192| = 192 × 60|| = 11520|| Therefore 3.2° = 11520|| CENTESIMAL SYSTEM (OR) FRENCH SYSTEM It is a system of measurement of angles in gradians. If a right angle is divided into 100 equal parts then each part is called a gradian. It is written as 100° 1 right angle = 100g Further, if one grade is divided into 100 equal parts, each part is called one minute. It is written as 100|
CLASS - X MATHEMATICS 3 FOUNDATION 8. INTRODUCTION TO TRIGONOMETRY VOL -II 1 grade = 100 minutes = 100| Again, if one minute is further divided into 100 equal parts, each is part is called one second and is written as 100|| 1 minute = 100 seconds = 100|| Examples i) It is a system of measurement of angles in gradians. 1° = 100 90 = 10 9 g g             45° = 45× 10 9 = 50 g  g      ii) Express 20g in sexagesimal system. We know 100g = 90° 1 = 90 100 = 9 10 g 10 o             20 = 9 10 × 20 =18° g CIRCULAR SYSTEM (OR) UNIVERSALLY ACCEPTD SYSTEM Consider a circle with centre O and radius r. Take an arc of a circle whose length is same as the radius (r) of the circle. Then what is an angle subtended at its centre? It is called a Radian. A radian is defined as the angle subtended by an arc of length equal to the radius of the circle at its center. It is written as 1C. It is also called circular measure. Note: Radian is not dependent on the radius of the circle. As it is accepted universally it has been used in various fields. It is called universally accepted system. 1.3 RELATION BETWEEN DEGREE AND RADIAN We know that if r is the radius of a circle, then the circumference (C) of the circle is given by 2πr. For one complete revolution, The length of the arc = circumference of circle = 2πr. l = 2πr [l is the length of arc] and rθ = 2πr [l = rθ] θ π = 2 r r = 2π radians ......(i) The circumference subtends at the centre an angle whose measure is 360°. θ = 3600 ...... (ii) (1) = (2) ⇒2π radians = 360° ⇒ π radians = 1800 1 radian = 180 ≠ = 180 22 / 7 = 7 180 22 × = 57° 17| 44.8|| . Approximately 57 17 0 | ∴ 1 radian = 1 =c 180 π where = 22 7 π       1c =57 16 0 | ,1° = 180 radians =1746 radian ≠ 1° = 0.01746c .