MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Maths), B.Ed, M.Phil (Maths) RAM RAJYA MRE, SIWAN XI th, XII th, TARGET IIT-JEE (MAIN + ADVANCE) & CMPATETIVE EXAM FR XI (PQRS) TRIGNMETRIC RATI AND IDENTITIES & Their Properties CNTENTS Ke Concept - I... Exericies-I... Exericies-II... Exericies-III... Solution Exercise Page...
THINGS T REMEMBER Measure of Angles : The measure of angle is the amount of rotation from the direction of one ra of the angle to the other. The initial and final positions of the revolving ra are respectivel called the initial and terminal sides (arms). Also, the revolving line is called the generating side. eg, if initial and final positions of the revolving ra are P and Q, then the angle formed will be PQ. If the rotation is in clockwise sense, the angle measured is negative and if the rotation is in anti-clockwise sense, the angle measured is positive. Q Sstem of Measurement of Angles : There are three sstem of mesurement (i) (ii) Sexagesimal Sstem Centesimal Sstem (iii) Circular sstem (i) (ii) Sexagesimal Sstem : In this sstem each angle is divided into 90 equal parts and each part is known as a degree. Thus a right angle is equal to 90 degree. ne degree is denoted as o. Each degree is divided into 60 equal parts each of which is known as one minute. ne minute is denoted as. Each minute is consist of 60 parts, each part is known as a second. ne second is denoted b. Hence, right angle = 90 o (90 degree) degree = 60 (60 minute) minute = 60 (60 second) Centesimal Sstem : In this sstem each angle is divied into 00 equal parts and one part is known as a grade. Thus one right angle is equal to 00 grade. ne grade is denoted as g. ne grade is divided into 00 equal parts, one part is known as a minute and is denoted as. ne minute is also divided into 00 euqal parts, one part is known as a second which is denoted b. Hence, right angle = 00 g (00 grade) grade = g = 00 (00 minute) minute = = 00 (00 second) (iii) Circular sstem : If the angle subtended b an arc of length l to the center of circle of radius r, is then l r P M.Sc. (Maths), B.Ed, M.Phil (Maths) r r l
If the length of arc is equal to the radius of the circle, then the angle subtended at the center of the circle will be one raidan. ne radian is denoted b c. The ratio of the circumference of the circle to the diameter of the circle is denoted b a greek letter and it is a constant quentit. Circumference of circle Diameter of circle This constant quantit is a irrational quentit and generall its approximate value is 7 and its general value upto six place of dicimal is 3.4857. Relation Among Degree, Radian and Grade : The number of radians of an angle subtended b an arc of a circle at the center is equal to the ratios of arc and radius. 80 o = c = 00 g and radian = 57 o 7 44.8 Trigonometric Ratios : Let us take a right angled triangle A right anled B. C A B Let C =, then sin Cosec sin cos sec cos tan cot tan Except these isx ratios, there are two other terms which are known as versed sine and coversed sine respectivel. ie, ver sin = - cos and cover sin = - sin Fundamental Relation Among Trigonometric Ratios : It is clear from the definitions of trigonometric rations that 3
. cosec sin cosec sin. sec cos sec cos 3. cot tan cot tan Representation of a Trigonometric Ratio in An ther Trigonometric Ratios : sin cos tan cot sec cosec tan sin sin cos tan cot cos sin cos tan cot sin tan sin cot sin sin sec sin cosec sin cos cos cos cos cos cos tan tan tan tan tan 4. sin tan and cos 5. cos + sin = 6. (a) + tan = sec (b) sec - tan = 7. (a) + cot = cosec (b) cosec - cot = cot cot sec sec sec cosec cos ec cosec cosec sec cot sec cos ec cot cot cos cot sin cot sec sec cosec sec cosec cosec Complementar and Supplementar Angles : If the sum of two angles is equal to right angle, then these angles are known as complementar angles of each other. Thus, and 90 o - are complementar angles of each other. Now, if the sum of two angles is equal to two right angles, then these angles are known as supplementr angles of each other. Thus, and 80 o - are supplementar angles of each other. eg, 3 o and 67 o are complementar angles of each other while 67 o and 3 o are supplementar angles of each other. Trigonometric Ratio of Complementar and supplementar Angles : sin cos tan cot sec cosec - -sin cos -tan -cot sec -cosec 90 o - cos sin cos tan cosec sin 90 o + cos -sin -cot -tan -cosec sec 80 o - sin -cos -tan -cot -sec cosec 80 o + -sin -cos tan cot -sec -cosec 4
Graph of Trigonometric Function :. Graph of sin x 0 II I Except sin and cosec, all ratios are negative. All trigonometric ratios are positive x x III IV Except tan and cot, all ratios are negative. Except cos and sec, all ratios are negative. 3, 3, B 3 = x - 3 - Facts Related to sin x. (a) Domain = R (b) Range = [-,] (c) Period =, (d) Graph of sin x is continuous for all real values of x. A h 3 3, h C x = -. Graph of cos x. (-, ) (0, ) (, ) D = x - 3 - (-, ) (, ) 3 x = - Facts Related to cos x. (a) Domain = R (b) Range = [-,] (c) Period = (d) Graph of cos x is continuous for all real values of x. 5
3. Graph of tan x. x 3 - - 3 x Facts Related to tan x. (a) Domain = R ~ (n + ), n I (b) Range = [-, ] (c) Period = (d) Graph of tan x is discontinuous at x = m, where m is an, odd integer. 4. Graph of cot x. Facts Related to tan x. (a) Domain = R ~ n, n I (b) Range = [-, ] (c) Period = (d) Graph of cot x is discontinuous at x = m, where m is an integer. 5. Graph of sec x. 6
Facts Related to sec x. (a) Domain = R ~ (n + ), n I (b) Range = (-, -][, ) (c) Period = (d) Graph of sec x is discontinuous at x = m, where m is an, odd integer. 5. Graph of cosec x. Facts Related to cosec x. (a) Domain = R ~ n, n I (b) Range = (-, -][, ) (c) Period = (d) Graph of cosec x is discontinuous at x = m, where m is an integer. Note : If measure of an angle is given in degree, then to convert it into radians, multipl the measure of an angle b o and if the measure of an angle is given in radians, then to convert it into degree, write 80 80 o at the place of sin (n + (-) n ) = sin, n I cos (n + ) = cos, n I sin (n + ) = tan, n I 7