Ch 03 FINAL 02.01 - NCERT

1 VA mathematician knows how to solve a problem,he can not solve it. MILNE IntroductionThe word trigonometry is derived from the Greek words trigon and metron and it means measuring the sides ofa triangle . The subject was originally developed to solvegeometric problems involving triangles. It was studied bysea captains for navigation, surveyor to map out the newlands, by engineers and others. Currently, trigonometry isused in many areas such as the science of seismology,designing electric circuits, describing the state of an atom,predicting the heights of tides in the ocean, analysing amusical tone and in many other earlier classes, we have studied the trigonometricratios of acute angles as the ratio of the sides of a rightangled triangle.

2 We have also studied the trigonometric identities and application oftrigonometric ratios in solving the problems related to heights and distances. In thisChapter, we will generalise the concept of trigonometric ratios to trigonometric functionsand study their AnglesAngle is a measure of rotation of a given ray about its initial point. The original ray isChapter3 TRIGONOMETRIC FUNCTIONSArya Bhatt (476-550)Fig the initial side and the FINAL position of the ray after rotation is called theterminal side of the angle. The point of rotation is called the vertex. If the direction ofrotation is anticlockwise, the angle is said to be positive and if the direction of rotationis clockwise, then the angle is negative (Fig ).

3 The measure of an angle is the amount ofrotation performed to get the terminal side fromthe initial side. There are several units formeasuring angles. The definition of an anglesuggests a unit, viz. one complete revolution from the position of the initial side asindicated in Fig is often convenient for large angles. For example, we can say that a rapidlyspinning wheel is making an angle of say 15 revolution per second. We shall describetwo other units of measurement of an angle which are most commonly used, measure and radian Degree measure If a rotation from the initial side to terminal side is th1360 ofa revolution, the angle is said to have a measure of one degree, written as 1.

4 A degree isdivided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree iscalled a minute, written as 1 , and one sixtieth of a minute is called a second, written as 1 .Thus,1 = 60 ,1 = 60 Some of the angles whose measures are 360 ,180 , 270 , 420 , 30 , 420 areshown in Fig FUNCTIONS Radian measure There is another unit for measurement of an angle, calledthe radian measure. Angle subtended at the centre by an arc of length 1 unit in aunit circle (circle of radius 1 unit) is said to have a measure of 1 radian. In the (i) to (iv), OA is the initial side and OB is the terminal side. The figures show theangles whose measures are 1 radian, 1 radian, 112 radian and 112 radian.

5 (i)(ii)(iii)Fig (i) to (iv)(iv)We know that the circumference of a circle of radius 1 unit is 2 . Thus, onecomplete revolution of the initial side subtends an angle of 2 generally, in a circle of radius r, an arc of length r will subtend an angle of1 radian. It is well-known that equal arcs of a circle subtend equal angle at the in a circle of radius r, an arc of length r subtends an angle whose measure is 1radian, an arc of length l will subtend an angle whose measure is lr radian. Thus, if ina circle of radius r, an arc of length l subtends an angle radian at the centre, we have = lr or l = r . Relation between radian and real numbersConsider the unit circle with centre O. Let A be any pointon the circle.

6 Consider OA as initial side of an the length of an arc of the circle will give the radianmeasure of the angle which the arc will subtend at thecentre of the circle. Consider the line PAQ which istangent to the circle at A. Let the point A represent thereal number zero, AP represents positive real number andAQ represents negative real numbers (Fig ). If werope the line AP in the anticlockwise direction along thecircle, and AQ in the clockwise direction, then every realnumber will correspond to a radian measure andconversely. Thus, radian measures and real numbers canbe considered as one and the Relation between degree and radian Since a circle subtends at the centrean angle whose radian measure is 2 and its degree measure is 360 , it follows that2 radian = 360 or radian = 180 The above relation enables us to express a radian measure in terms of degreemeasure and a degree measure in terms of radian measure.

7 Using approximate valueof as 227, we have1 radian = 180 = 57 16 = 180 radian = radian relation between degree measures and radian measure of some common anglesare given in the following table:AO1P12 1 2Q0 Fig 45 60 90 180 270 360 Radian 6 4 3 2 3 22 2021-22 TRIGONOMETRIC FUNCTIONS 53 Notational ConventionSince angles are measured either in degrees or in radians, we adopt the conventionthat whenever we write angle , we mean the angle whose degree measure is andwhenever we write angle , we mean the angle whose radian measure is .Note that when an angle is expressed in radians, the word radian is frequentlyomitted. Thus, 180 and454= = are written with the understanding that and 4are radian measures.

8 Thus, we can say thatRadian measure = 180 Degree measureDegree measure = 180 Radian measureExample 1 Convert 40 20 into radian We know that 180 = 20 = 40 13 degree = 180 1213 radian = 121 540 20 = 121 540 2 Convert 6 radians into degree We know that radian = 180 .Hence 6 radians= 180 6 degree= 1080722 degree= 343711degree= 343 + 76011 minute[as 1 = 60 ]= 343 + 38 + 211 minute[as 1 = 60 ]= 343 + 38 + = 343 38 11 6 radians = 343 38 11 3 Find the radius of the circle in which a central angle of 60 intercepts anarc of length cm (use 22 7=).2021-2254 MATHEMATICSS olution Here l = cm and = 60 = 60 radian =1803 Hence,by r = l, we haver = 3 7= 22 = cmExample 4 The minute hand of a watch is cm long.

9 How far does its tip move in40 minutes? (Use = ).Solution In 60 minutes, the minute hand of a watch completes one revolution. Therefore,in 40 minutes, the minute hand turns through 23 of a revolution. Therefore, 2 = 360 3or 4 3 radian. Hence, the required distance travelled is given by l =r = 4 3cm = 2 cm = 2 cm = 5 If the arcs of the same lengths in two circles subtend angles 65 and 110 at the centre, find the ratio of their Let r1 and r2 be the radii of the two circles. Given that 1 = 65 = 65180 = 13 36 radianand 2 = 110 = 110180 = 22 36radianLet l be the length of each of the arc. Then l = r1 1 = r2 2, which gives13 36 r1 = 22 36 r2 , , 12rr= 2213 Hence r1 : r2 = 22 : the radian measures corresponding to the following degree measures:(i) 25 (ii) 47 30 (iii) 240 (iv) 520 2021-22 TRIGONOMETRIC FUNCTIONS the degree measures corresponding to the following radian measures(Use 22 7=).

10 (i)1116(ii) 4(iii)5 3(iv)7 wheel makes 360 revolutions in one minute. Through how many radians doesit turn in one second? the degree measure of the angle subtended at the centre of a circle ofradius 100 cm by an arc of length 22 cm (Use 22 7=). a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length ofminor arc of the in two circles, arcs of the same length subtend angles 60 and 75 at thecentre, find the ratio of their the angle in radian through which a pendulum swings if its length is 75 cmand the tip describes an arc of length(i)10 cm(ii)15 cm(iii)21 Trigonometric FunctionsIn earlier classes, we have studied trigonometric ratios for acute angles as the ratio ofsides of a right angled triangle.