Title tri notes (2).pdf ✅

DEPARTMENT OF COLLEGIATE AND TECHNICAL EDUCATION GOVERNMENT POLYTECHNIC,RAICHUR 2020 ENGINEERING MATHEMATICS UINT-3, TRIGONOMETRY STUDY MATERIAL Prepared by: Ramachandra Sutar N E A R G O V T I T I C O L L E G E A M A R K H E D L A Y O U T R A I C H U R - 5 8 4 1 0 3
Engineering Mathematics_Unit-3, Trigonometry_Ramachandra Sutar Page 2 UNIT-3.TRIGONOMETRY 3.1 INTRODUCTION: The literal meaning of the word trigonometry is the ‘science of triangle measurement’. The word ‘trigonometry’ is derived from two greek words ‘trigon’ and ‘metron’ which means measuring the sides of a triangle. It had its beginning more than two thousand years ago as a tool for astronomers. The Babylonians, Egyptians, Greeks and the Indians studied trigonometry only because it helped them in unraveling the mysteries of the universe. In modern times, it has gained wider meaning and scope. Presently, it is defined as that branch of mathematics which deals with the measurement of angles, whether of triangle or any other figures. At present trigonometry is used in surveying, astronomy, navigation, physics, engineering etc. [Trigon → 3 angles/ sides, Metry → Measurement  Trigonometry → Measurement of triangle] 3.2 RECAPITULATION OF TRIGONOMETRY [Studied in Previous Classes] 1. MEASUREMENT OF ANGLES – RADIAN MEASURES Angle: An angle is a figure obtained by rotating a ray about its initial point. If a ray OA is rotated about its initial point O and reaches to final position OB then the figure obtained is called an angle AOB and it is denoted by AOB .Here OA is called initial line, OB is called terminal line and O is called the vertex of an angle. An angle has positive measure if it is measure in anti-clockwise direction; otherwise it is a negative measure. An angle can be measured in any one of the following systems: 1. Sexagesimal system: In this system a right angle is divided into 90 equal parts; each part is equal to one degree. A degree is divided into 60 equal parts; each part is equal to one minute. A minute is divided into 60 equal parts; each part is equal to one second. A right angle = 90o , 1/ = 60I , 1’ = 60//,Unit= Degree 2. Centesimal system: In this system a right angle is divided into 100 equal parts; each part is equal to one grade. A grade is divided into 100 equal parts; each part is equal to one minute. A minute is divided into 100 equal parts; each part is equal to one second. A right angle = 100g , 1g = 100/ , 1’ = 100//, Unit=Grade 3. Radian or Circular measure: “A radian is the angle subtended at the centre of the circle by an arc whose length is equal to the radius of the circle”. A right angle = 2 c  radian  90o = 100g = 2 c  A B O Angle Initial line Terminal line
Engineering Mathematics_Unit-3, Trigonometry_Ramachandra Sutar Page 3 Theorem-1: A radian is a constant angle or  radians = 180o ( 2  radians = 90o ) A radian is less than 60o or 1 radian = 57o 17/ 45// Theorem-2: If  is the angle subtended at the centre of the circle by an arc of length S then S = r S = Arc length r = Radius of Circle  = Angle in radian Theorem-3: Area of the sector is A = 2 1 r 2 where A = Area of sector Also, A= rs 2 1 r = Radius of Circle  = Angle in radian Radian measure of some common angles Degree 15o 30o 45o 60o 75o 90o 105o 120o 138o 180o 165o 180o 270o 360o Radians 12  6  4  3  12 5 2  12 7 3 2 4 3 6 5 12 11  2 3 2 Conversions: i) To convert degree into radians multiply by 180   x o = x 180   radian ii) To convert radians into degrees multiply by  180  y c = y  180  degree Ex: (i) 1200 , 241 , 2411 = 123o + 3600 24 60 24o o + = 1230 + 0.40 + 0.00670 = 23.40670 = 123.4067 x 180 c  = 2.148c 2. TRIGNOMETRIC RATIOS The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Let us see how are these ratios or functions, evaluated in case of a right-angled triangle.
Engineering Mathematics_Unit-3, Trigonometry_Ramachandra Sutar Page 4 Consider a right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse are referred to as the adjacent and opposite sides. The six important trigonometric functions (trigonometric ratios) are calculated using the below formulas and considering the above figure. It is necessary to get knowledge about the sides of the right angled triangle because it defines the set of important trigonometric functions. Functions Abbreviation Relationship to sides of a right angled triangle Sine Function sin Opposite side/ Hypotenuse Cosine Function cos Adjacent side / Hypotenuse Tangent Function tan Opposite side / Adjacent side Cotangent Function cot Adjacent side / Opposite side Secant Function sec Hypotenuse / Adjacent side Cosecant Function cosec Hypotenuse / Opposite side i.e (i) Sin = r y Hyp Opp = (ii) Cos = r x Hyp. Adj. = (iii) tan = x y Adj. Opp. = (iv) Cot = y x Opp. Adj. = (v) Sec = x r Adj. Hyp. = (vi) Cosec = y r Opp. Hyp. = Reciprocal relations: (i) sin = cos ec 1 or cosec = sin  1 or sin.cosec=1 (ii) cos = sec  1 or sec = cos  1 or cos.sec=1 (iii) tan = cot  1 or cot = tan 1 or tan.cot=1 Quotient Relations: (i) tan =   cos sin (ii) cot =   sin cos C A B Adjacent Side=x Hypotenuse= r Opposite Side=y