TRIGONOMETRIC FUNCTIONS

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1 TRIGNMETRIC FUNCTINS INTRDUCTIN In general, there are two approaches to trigonometry ne approach centres around the study of triangles to which you have already been introduced in high school ther one is the unit circle approach in which we use radian measure of an angle to define trigonometric functions of real numbers It meets the requirements of calculus and modern mathematics ANGLE AND ITS MEASUREMENT What is an angle? C An angle is made up of two rays with a common end point This end point is the verte of the angle The rays Angle are the sides of the angle In fig, the angle may be named ABC or B B A Fig Signs of angles The above definition is useful in geometry In trigonometry, we need broader definition of an angle Let a revolving ray starting from X, rotate about in a plane and stop at position P Then it is said to trace out an angle XP X is called initial side, P is final or terminal side and is the verte of the angle If rotation is anticlockwise, the angle is positive, if rotation is clockwise, the angle is negative An angle is said to lie in a particular quadrant if the terminal side of the angle lies in that quadrant Terminal side Initial side Two angles are called coterminal if they have the same position of initial side and terminal side Thus, keeping the initial side fied (as X), there are an unlimited number of angles corresponding to each ray Initial side Fig X Measuring angles The measure of an angle is the amount of rotation made to get the terminal side from its initial side There are several units for measuring angles θ P +ve angle X Terminal side Fig Terminal side Initial side θ ve angle P P X

2 94 MATHEMATICS XI ne unit of measuring angle is one complete rotation (or revolution) as shown in fig 4 This unit is convenient for large angles For eample, we can say that a rapidly spinning wheel of a machine is making an angle of 0 revolutions per second The most commonly used units of measurements are : Initial side Terminal side Fig 4 X Degree measure In this system an angle is measured in degrees, minutes and seconds A complete rotation describes 60 ie th of a complete rotation 60 right angle 90 (Since right angle is th of full rotation) 4 A degree is further subdivided as degree 60 minutes, written as 60 and minute 60 seconds, written as 60 Radian measure In this system an angle is measured in radians A radian is an angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle In fig, let AB be an arc of a circle with centre and of radius r such that length of arc AB r, then c r r A AB radian (written at c ) Fig Theorem A radian is a constant angle Proof In fig, let AB be an arc of a circle with centre and of radius r such that length of arc AB r Then, by definition, AB radian We shall use our knowledge from geometry that angles at the centre of a circle are in the ratio of subtending arcs As the full circumference subtends an angle of 60 at the centre, B AB 60 arc AB circumference r r AB 60 radian 80 Since the right hand side is independent of radius r, we find that a radian is a constant angle Radian (circular) measure of an angle The radian (circular) measure of an angle is the number of radians it contains Corollary radians 80 right angles Relation between degree and radian From the above theorem, we know that radians 80 This gives us formula of conversion from one system to other Taking ~, we get radian right angles

3 TRIGNMETRIC FUNCTINS ie radian nearly Also 80 radians 4 Notational convention radians 0074 radians nearly 80 If an angle is given without mentioning units, it is assumed to be in radians Thus, whenever we write angle θ, we mean the angle whose degree measure is θ and whenever we write angle, we mean the angle whose radian measure is Hence 80 and 60 are written with the assumption that and are radian measures The relation between degree measures and radian measures of some standard angles are given below : Degrees Radians Length of an arc of a circle Theorem If an arc of length l subtends an angle θ radians at the centre of a circle of radius r, then l rθ Proof Let arc AP of length l subtend an angle θ radians at the centre Mark point B on circumference such that AB radian Thus length of arc AB r Now AP AB length of arc AP length of arc AB P θ c B r l A θ radians radian l r l rθ Fig 6 NTE It is assumed that l and r have same linear units ILLUSTRATIVE EXAMPLES Eample Draw diagrams for the following angles In which quadrant do they lie? (i) (ii) 740 Solution Diagrams are given below for the two angles X is initial side and P is terminal side From the diagram, we see that lies in second quadrant and lies in fourth quadrant P Y Y X X X X P Y Fig 7 Y

4 96 MATHEMATICS XI Eample Convert the following into radian measures : (i) (NCERT) (ii) 47 0 (NCERT) (iii) 7 0 Solution We know that 80 radians, therefore, 80 radians (i) 80 radians radians 6 0 (ii) radians 9 7 radians 0 (iii) radians radians Eample Convert the following radian measures into degree measures use (i) 6 (ii) 4 (iii) 7 6 (NCERT) Solution We know that radians 80, therefore, radian 80 7 : (i) radians (ii) 4 radians (approimately) 9 7 (approimately) 60 (iii) 7 radians Eample 4 Epress in radians the fourth angle of a quadrilateral which has three angles , and 9 Take Solution The sum of three given angles ( Q 90 0 and 84 4 )

5 TRIGNMETRIC FUNCTINS 97 As the sum of all four angles of a quadrilateral is 60, the fourth angle 60 (4 4 0 ) To convert it into radians : 4 0 ( Q ) radians ( Q 80 radians) 7 0 radians 80 radians nearly Eample Find the angle in radian through which a pendulum swings if its length is 7 cm and the tip describes an arc of length cm (NCERT) Solution The pendulum describes a circle of radius 7 cm and its tip describes an arc of length cm Let θ radians be the angle through which the pendulum swings Here r 7 cm and l cm r θ l r 7 7 l Fig 8 Eample 6 Find the radius of the circle in which a central angle of 60 intercepts an arc of 74 cm length use (NCERT) 7 Solution Here l 74 cm and θ measure of θ is Let r cm be the radius of the circle radians radians, so the radian We know that θ l r r l θ Hence, the radius of the circle 7 cm Eample 7 Find the degree measure of the angle subtended at the centre of a circle of diameter 00 cm by an arc of length cm use (NCERT) 7 Solution Here radius of circle r diameter 00 cm 00 cm, length of arc l cm

6 98 MATHEMATICS XI θ l r radians 00 radians 0 80 degrees degrees 6 o Eample 8 In a circle of diameter 40 cm, the length of a chord is 0 cm Find the length of the minor arc of the circle (NCERT) Solution Here radius of circle r 40 cm 0 cm Let be the centre of circle and AB be a chord of length 0 cm Since A B 0 cm and AB 0 cm, ΔAB is equilateral, therefore, AB radians radians Let the length of the minor arc AB be l, then Fig 9 l r θ 0 cm 0 cm Eample 9 If the arcs of the same length in two circles subtend angles of 60 and 7 at their respective centres, find the ratio of their radii (NCERT) Solution Ler r and r be the radii of the two given circles and let their arcs of the same length, say l, subtend angles of 60 and 7 at respective centres Using the formula l r θ, we get c l r r Hence r : r : 4 c, 7 7 c 80 r r 4 Eample 0 The large hand of a big clock is cm long How many cm does its tip move in 9 minutes? Solution The angle traced by the large hand in 60 minutes 60 radians ( Q 80 c ) The angle traced by the large hand in 9 minutes 9 radians 60 0 radians Let l be the length of the arc moved by the tip of the minutes hand, then c l rθ cm cm cm Eample A wheel of a motor is rotating at 00 rpm If the radius of the wheel is cm, what linear distance does a point of its rim traverse in 0 seconds? What steps should be taken to discourage reckless driving? Solution Radius of the wheel cm, circumference of the wheel r cm 0 cm 7 (Value Based) Hence, the linear distance travelled by a point of the rim in one revolution 0 cm

7 TRIGNMETRIC FUNCTINS 99 Now, the speed of the wheel is 00 revolutions per minute second The number of revolutions in 0 seconds The linear distance travelled by a point of the rim in 0 seconds (600 0) cm 000 cm km ie 0 revolutions per Speed limits should be fied and monitored properly There should be fines and imprisonment for reckless driving Licences of drivers involved in reckless driving should be cancelled or suspended Conducting proper training of drivers should be mandutory to teach them about the risks associated Eample In a right angled triangle, the difference between two acute angles is 8 measure Epress the angles in degrees Solution Since the triangle is right angled, sum of two acute angles is 90 Let the two acute angles be and y, > y Then + y 90 in radian (i) Also y 8 radians 80 8 ie y 0 Solving (i) and (ii) simultaneously, we get 0, y 40 ( Q radians 80 ) (ii) Eample If the angles of a triangle are in the ratio : 4 :, find the smallest angle in degrees and the greatest angle in radians Solution Let the three angles be, 4 and degrees, then The smallest angle degrees degrees 4 and the greatest angle degrees degrees radians radians Eample 4 The angles of a triangle are in AP and the number of degrees in the least to the number of radians in the greatest is 60 : Find the angles in degrees and radians Solution Let the angles be (a d ), a, (a + d ), where d > 0 Then (a d ) + a + (a + d ) 80 a 80 a 60 Hence the angles are (60 d), 60, (60 + d ) Least angle (60 d ) Greatest angle (60 + d ) (60 + d) By given condition, (60 d) : (60 + d) ( 60 d) ( 60 d) ( 60 + d) 60 + d 80 radians : 60 + d 80 d 4d 0 d 0 (As 80 radians radians) 80

8 00 MATHEMATICS XI Thus the angles are (60 0), 60, (60 + 0) ie 0, 60, 90 In radians, the angles are 0 80, 60 80, ie,, radians 6 Eample Taking the moon s distance from the earth as km and the angle subtended by the moon at any point on the earth as half a degree, estimate the diameter of the moon (Use 46) Solution As arc AB is a part of very large circle (of radius km), the diameter AB of the moon is approimately equal to the length of the arc AB Now, angle θ radians radians AB r θ km 000 km km 46 km EXERCISE Very short answer type questions ( to 4) : Draw diagrams for the following angles : (i) (ii) 740 In which quadrant do they lie? (iii) Find another positive angle whose initial and final sides are same as that of, and indicate on the same diagram If θ lies in second quadrant, in which quadrant the following will lie? (i) θ (ii) θ (iii) θ Epress the following angles in radian measure : (i) 40 (ii) (iii) 70 4 Epress the following angles in degree measure : (i) (ii) 4 (iii) 4 Epress the following angles in radian measure : (i) (ii) 0 (iii) 40 0 (iv) Find the degree measures corresponding to the following radian measures : (i) 6 (ii) (iii) 4 7 A wheel makes 60 revolutions in a minute Through how many radians does it turn in one second? (NCERT) 8 Find the angle in radians through which a pendulum swings if its length is 7 cm and the tip describes an arc of length : (i) 0 cm (ii) cm (NCERT) 9 Find the radius of the circle in which a central angle of 4 makes an arc of length 87 cm use 7 0 Find the length of an arc of a circle of diameter 0 cm which subtends an angle of 4 at the centre θ r B Fig 0 A

9 TRIGNMETRIC FUNCTINS 0 An engine is travelling along a circular railway track of radius 00 metres with a speed of 60 km/hr Find the angle in degrees turned by the engine in 0 seconds What role does railways play in India s transportation system especially for goods? (Value Based) If the arcs of the same length in two circles subtend angles of 6 and 0 at their respective centres, find the ratio of their radii Large hand of a clock is cm long How much distance does its etremity move in 0 minutes? 4 The minute hand of a watch is cm long How far does its tip move in 40 minutes? Use 4 Find the angles in degrees through which a pendulum swings if its length is 0 cm and the tip describes an arc of length : (i) 0 cm (ii) 6 cm (iii) 6 cm use 7 6 Find the length of an arc of a circle of radius 7 cm that spans a central angle of measure 6 Take 46 7 The circular measures of two angles of a triangle are and Find the third angle in degree measure Take 7 8 The difference between two acute angles of a right angled triangle is in radian measure Find these angles in degrees 9 The angles of a triangle are in AP and the greatest angle is double the least Find all the angles in circular measure 0 Estimate the diameter of the sun supposing that it subtends an angle of at the eye of an observer Given that the distance of the sun is km Take 7 TRIGNMETRIC FUNCTINS F A REAL NUMBER In calculus and in many applications of mathematics, we need the trigonometric functions of real numbers rather than angles By making a small but crucial change in our viewpoint, we can define trigonometric functions of real numbers In the previous section of this chapter, we found that the radian measure of an angle is given by the equation θ l In this result, we assumed that l and r have same linear units and therefore r the ratio l r is a real number with no units In particular, in the equation θ l r, if we take r then we get θ l l (a real number) Consider the unit circle ie a circle of radius unit (in length) with centre Let A be any point on the circle P Consider A as the initial side of the angle AP, then the radian measure of the angle AP is equal to the length of the arc AP (see fig ) Here, we have used the letter, rather than our usual θ, to emphasize the fact that both the radian measure of the angle and the measure of the arc AP are given by the same real number Fig A The conventions regarding the measure of arc length on the unit circle are same as those for angles In fig, we measure arc length (or radian measure of the angle) from the point A and take positive in anticlockwise direction and negative in the clockwise direction

10 0 MATHEMATICS XI P A A P (i ) is positive (ii ) is negative Fig You may think of as either the measure of an arc length or the radian measure of an angle But in both cases, is a real number Trigonometric (or circular) functions of a real number Let be the centre of a circle of unit radius Choose the aes as shown in fig Let A be the point (, 0) and P(a, b) a point on the unit circle such that the length of arc AP is equal to, or equivalently, let P (a, b) be the point where the terminal side of the angle AP with radian measure meets the unit circle, then the two basic trigonometric (or circular) functions of the real number are defined as (i) sin b, for all R (ii) cos a, for all R REMARK Note that sin AP MP P b sin etc Hence we do not distinguish between trigonometric ratios of an angle AP whose radian measure is and the trigonometric function of a real number From the above definitions it follows that if P is a point on the unit circle such that length of arc AP is or equivalently P is a point where the terminal side of the angle with radian measure meets the unit circle, then the co-ordinates of the point P are (cos, sin ) Values of sin and cos at 0,,,, We know that in unit circle, the length of circumference is If we start from A and move in the anticlockwise direction then at the points A, B, A, B and A, the arc lengths travelled are 0,,, and Also the co-ordinates of the points A, B, A, B and A are (, 0), (0, ), (, 0), (0, ) and (, 0) respectively Therefore, (i) sin 0 0 (ii) cos 0 (iii) sin (iv) cos 0 X X X A (, 0) Y Y Fig Y Y Fig 4 Y Y M Fig B(0, ) B (0, ) P (a, b) A X (, 0) Unit circle P (a, b) A P(a, b) X A(, 0)X

11 TRIGNMETRIC FUNCTINS 0 (v) sin 0 (vi) cos (vii) sin (viii) cos 0 (i) sin 0 () cos Further, sin 0 when the point P on the unit circle coincides with the points A or A ie when 0,,,, or,,, ie when 0, ±, ±, ie when is an integral multiple of ie when n where n is any integer Also cos 0 when the point P on the unit circle coincides with the points B or B ie when,,,, or,,, ie when ±, ±, ±, ie when is an odd multiple of ie when (n + ) where n is any integer Thus, sin 0 when n, n is any integer and cos 0 when (n + ), n is any integer ther trigonometric functions The other trigonometric functions of the real number are defined in terms of sine and cosine functions as follows : cosec sec, n, n is any integer sin cos tan sin cos, (n + ), n is any integer, (n + ), n is any integer cot cos, n, n is any integer sin 4 Relations between trigonometric functions of real numbers The following identities are the immediate consequences of the above definitions of trigonometric functions : Reciprocal relations (i) sin cosec and cosec sin (ii) cos sec and sec cos (iii) tan cot and cot tan From these results, it follows that : (i) sin cosec (ii) cos sec (iii) tan cot Quotient relations (i) tan sin cos (ii) cot cos sin Fundamental identity sin + cos for all R Proof Since the point P(a, b) lies on the unit circle (see fig ), with centre (0, 0), we have P ( a 0) + ( b 0) a + b

12 04 MATHEMATICS XI Now replacing a by cos and b by sin, we get cos + sin Thus, sin + cos for all R Two other ways of writing this identity are : sin cos and cos sin ther fundamental identities (i) + tan sec, (n + ), n is any integer (ii) + cot cosec, n, n is any integer Proof We know that sin + cos, for all R (i) Dividing both sides of the identity sin + cos by cos, we get sin cos cos +, assuming that cos 0 cos cos tan + sec, (n + ), n is any integer Two other ways of writing this identity are : sec tan and sec tan (ii) Dividing both sides of the identity sin + cos by sin, we get sin cos +, assuming that sin 0 sin sin sin + cot cosec, n, n is any integer Two other ways of writing this identity are : cosec cot and cosec cot 6 pposite Real Number Identities (i) sin ( ) sin (ii) cos ( ) cos (iii) tan ( ) tan (iv) cot ( ) cot (v) sec ( ) sec (vi) cosec ( ) cosec Proof Let be the centre of a unit circle and A be the point (, 0) Let P be a point on the unit circle such Y that length of arc AP is equal to (or equivalently, let P be the point where the terminal side of the angle P with radian measure meets the unit circle), then the co-ordinates of the point P are (cos, sin ) n the other hand, if we start from A and move on X M A(, 0) X the unit circle in the clockwise direction to the point Q such that arc length AQ, the co-ordinates of the point Q are (cos ( ), sin ( )) Q Y Let PQ meet A at M In Δs PM and QM, Fig 6 P Q, M M and PM QM ( Q length of arc AP length of arc AQ, so these subtend equal angles at the centre of the circle) ΔPM ΔQM MP MQ and MP 90

13 TRIGNMETRIC FUNCTINS 0 Hence, the -coordinates of the points P and Q are same, while the y-coordinates are negatives of each other Thus, we have cos ( ) cos and sin ( ) sin To establish the third identity, we have tan ( ) sin ( ) sin tan cos ( ) cos We leave the proofs of the other three identities for the reader REMARK In the above figure, the arc length terminates in the first quadrant However, the argument used will work no matter where the arc length terminates 7 Periodic functions Definition A function f is said to be periodic iff there eists a constant real quantity p such that f ( + p) f () for all D f There may eist more than one value of p satisfying the above relation The least positive value of p satisfying above relation is called the period of f Periodicity of sine and cosine functions Let be the centre of a unit circle and A be the point (, 0) Let P be a point on the unit circle such that length of arc AP is equal to We know that the circumference of the unit circle is ( Q circumference r, here r ) Thus, if we begin from any point P on the unit circle and travel a distance of along the perimeter, we return to the same point P In other words, the arc lengths and + (measured from A, as usual) yield the same terminal point P on the unit circle Since the trigonometric functions are defined in terms of the co-ordinates of the point P, we have sin ( + ) sin, cos ( + ) cos These results are true for all real values of They provide us useful information about their graphs; the graphs of both functions repeat themselves at intervals of Further, as these functions do not change on changing to +, therefore, sine and cosine functions are periodic with period Similar results hold for other trigonometric functions in their respective domains : tan ( + ) tan, cot ( + ) cot sec ( + ) sec, cosec ( + ) cosec As these functions do not change on changing to +, therefore, all these functions are periodic The period of secant and cosecant functions is ; and the period of tangent and cotangent functions is (see article 4) The above results can be generalised If we begin from any point P on the unit circle and make two complete anticlockwise revolutions, the arc length travelled is ( ) ie 4 For three complete revolutions, the arc length travelled is ( ) ie 6 In general, if n is any integer, the arc length travelled for n complete revolutions is n (when n is positive, the revolution are anticlockwise; when n is negative, the revolutions are clockwise) Consequently, we get the following results : For any real number and any integer n, we have sin ( + n) sin, cos ( + n) cos X Y Y Fig 7 P + A(, 0) X

14 06 MATHEMATICS XI Similar results hold for other trigonometric functions in their respective domains : tan ( + n ) tan, cot ( + n ) cot, sec ( + n ) sec, cosec ( + n ) cosec 8 Values of trigonometric functions for 0, 6, 4,,,,, In our earlier classes, we found the values of trigonometric ratios for 0, 4 and 60 The values of trigonometric functions for, and are same as that of trigonometric ratios for 0, and 60 respectively The values of trigonometric functions for 0, 6, 4,,,, memorised with the help of following table : and can be sin cos tan 0 cot nd sec cosec nd ( nd stands for not defined ) 0 0 nd 0 nd 0 0 nd 0 nd nd nd nd nd 9 Signs of trigonometric functions Let be the centre of a unit circle and A be the point (, 0) Let P(a, b) be a point on the unit circle such that length of arc AP or equivalently, let P(a, b) be the point where the terminal side of the angle AP with radian measure meets the unit circle, then the si trigonometric functions of the real number are defined as (i) sin b, for all R (ii) cos a, for all R (iii) tan b a, (n + ), n is any integer (iv) cot a, n, n is any integer b II X A III Y B B Y Fig 8 P(a, b) I A(, 0) X IV (v) sec a, (n + ), n is any integer (vi) cosec, n, n is any integer b We know that the circumference of the unit circle is Note that in the unit circle, a and b

15 TRIGNMETRIC FUNCTINS 07 Also a > 0, b > 0 in I quadrant, a < 0, b > 0 in II quadrant, a < 0, b < 0 in III quadrant, a > 0, b < 0 in IV quadrant Signs of trigonometric functions of (a real number) In the first quadrant, a > 0, b > 0 Consequently, all the si trigonometrical functions are +ve In the second quadrant, a < 0, b > 0 So sin b and cosec b are positive and all other four trigonometric functions ie cos a, tan b a, cot a b and sec a are negative In the third quadrant, a < 0, b < 0 So tan b a and cot a b are positive and all other four trigonometric functions ie sin b, cos a, sec a and cosec b are negative In the fourth quadrant, a > 0, b < 0 So cos a and sec a are positive and all other four trigonometric functions ie sin b, tan b a, cot a b and cosec b This can be summarised as : are negative Quadrant I II III IV t-functions which All sin tan cos are + ve cosec cot sec This table can be memorised with the help of phrase : Add Sugar To Coffee All sin tan cos (I) (II) (III) (IV) 0 Domain and range of trigonometric functions Domain of trigonometric functions We know that sin and cos are defined for all real values of ; tan and sec are defined for all real values of ecept when (n + ), where n is an integer; cot and cosec are defined for all real values of ecept when n, where n is an integer This can be summarised as : Function Domain sin, cos tan, sec cot, cosec all real numbers all real numbers other than ( n + ), n Z all real numbers other than n, n Z Range of trigonometric functions As a and b in unit circle, cos and sin Thus, the maimum and minimum values of sin and cos are and respectively Since tan b a and cot a, and any of a and b (see fig 6) can be greater than the other, b tan and cot can take any real value

16 08 MATHEMATICS XI Now a, a 0 a or a sec or sec Also b, b 0 b or b cosec or cosec This information can be summarised as : Function Range sin, cos [, ] tan, cot any real value sec, cosec any real value ecept (, ) ILLUSTRATIVE EXAMPLES Eample Which of the si trigonometric functions are positive for 0? Solution Given 0 We know that terminal position of + n, where n Z, is the same as that of Here, 0 +, which lies in the second quadrant (This process of finding a coterminal angle or reference number results in a angle or number α, 0 α <, so that we can determine in which quadrant the given angle or number lies) Therefore, 0 lies in the second quadrant Hence sin and cosec are +ve while the other four trigonometric functions ie cos, tan, cot and sec are ve Eample If sin functions and lies in second quadrant, find the values of other five trigonometric (NCERT) Solution Given sin and lies in the second quadrant cosec sin We know that sin + cos cos sin 9 cos ± 4 6 But lies in the second quadrant and cos is ve in the second quadrant, therefore, cos 4 sec cos 4 Further, tan sin cos 4 cot 4 tan 4

17 TRIGNMETRIC FUNCTINS 09 Eample If tan and lies in the second quadrant, find the values of other five trigonometric functions (NCERT) Solution Given tan and lies in the second quadrant cot tan We know that sec + tan sec sec ± 44 But lies in the second quadrant and sec is ve in the second quadrant, therefore, sec cos sec Further, sin sin cos cos tan cos cosec sin Eample 4 If sec and lies in the fourth quadrant, find the values of other five trigonometric functions (NCERT) Solution Given sec and lies in the fourth quadrant cos sec We know that sin + cos sin cos sin ± But lies in the fourth quadrant and sin is ve in the fourth quadrant, therefore, sin cosec sin Further, tan sin cos cot tan Eample If sin, find the quadrant in which can lie Also find the values of remaining trigonometric functions of Solution Given sin which is +ve, therefore, can lie in first or second quadrant We know that sin + cos cos sin cos ± 69

18 0 MATHEMATICS XI Two cases arise : Case I When lies in first quadrant, cos is +ve cos sin, tan, cot cos tan, sec, cosec cos sin Case II When lies in second quadrant, cos is negative cos, tan sin, cos, sec cos, cot cosec tan sin Eample 6 If tan α, find the values of the remaining trigonometric functions of α Solution Given tan α which is ve, therefore, α lies in second or fourth quadrant Also sec α + tan α + ( ) sec α ± Two cases arise : Case I When α lies in the second quadrant, sec α is ve sec α cos α sin α sin α cos α tan α cos α cos α cosec α Also tan α cot α Case II When α lies in the fourth quadrant, sec α is + ve sec α cos α sin α sin α cos α cos α tan α cos α cosec α Also tan α cot α Eample 7 If cos and < <, find the value of 4 tan cosec Solution Given cos and < < ie lies in the third quadrant sec cos We know that sin + cos sin cos sin ±

19 TRIGNMETRIC FUNCTINS But lies in the third quadrant and sin is ve in the third quadrant, therefore, sin cosec sin Further, tan sin cos 4 tan cosec Eample 8 If lies in the second quadrant, then show that sin + + sin + sin sin sec (NCERT Eamplar Problems) Solution LHS sin + sin + + sin sin sin sin + sin + sin sin ( Q, for all R) cos cos (Given lies in second quadrant, so cos is ve cos cos ) cos sec RHS Eample 9 (i) If sec + tan p, obtain the values of sec, tan and sin in terms of p (ii) If p 4 in above case, then find sin and cos In which quadrant does lie? Solution (i) Given sec + tan p (i) We know that sec tan (sec + tan ) (sec tan ) p (sec tan ) (using (i)) sec tan p (ii) From (i) and (ii), we get sec p + p and tan p p sec p + p and tan p p Now tan sec sin cos sin cos sin tan sec ( p )/ p ( p + )/ p p p +

20 MATHEMATICS XI (ii) If p 4, we get sin p p p 4 8 cos sec p As both sin and cos are +ve, lies in the first quadrant Eample 0 If sin, find the value of sec tan sec + tan Solution Given sin sin Now sec sec + tan tan cos cos + sin cos sin cos sin + sin Eample Find the value of tan + cos 4 + sec cos Solution tan + cos 4 + sec cos ( ) (0) Eample Find the values of the following : (i) tan 9 (NCERT) (ii) sin (NCERT) (iii) cot 4 (NCERT) (iv) cosec 9 Solution (i) tan 9 tan 6 + tan + tan ( Q tan (n + ) tan ) (ii) sin sin 4 + sin ( ) + sin ( Q sin (n + ) sin ) (iii) cot 4 cot 4 + cot ( ) cot 4 ( Q cot (n + ) cot )

21 TRIGNMETRIC FUNCTINS (iv) cosec 9 cosec 9 ( Q cosec ( ) cosec ) cosec 6 + cosec + cosec Eample Find the values of the following : (i) cos ( 70 ) (NCERT) (ii) cosec ( 40 ) (NCERT) Solution (i) cos ( 70 ) cos ( ) cos ( ) cos ( ) + cos ( Q cos (n + ) cos ) 0 (ii) cosec ( 40 ) cosec ( ) cosec ( 4 ) + 6 cosec 6 ( Q cosec (n + ) cosec ) Eample 4 Is the equation sin cos possible? Solution sin cos ( cos ) cos cos cos cos + cos 6 0 ( cos ) (cos + ) 0 cos 0 or cos + 0 cos or cos, both of which are impossible as cos Hence, the equation sin cos is not possible Eample For what real values of is the equation cos θ + possible? Solution Given cos θ + cos θ + 0, which is a quadratic in As is real, discriminant 0 ( cos θ) 4 0 cos θ but cos θ cos θ cos θ, Case I When cos θ, we get + 0 Case II When cos θ, we get Hence, the values of are and

22 4 MATHEMATICS XI Eample 6 If A cos + sin 4 for all in R, then prove that 4 A Solution A cos + sin 4 cos + sin sin cos + sin (NCERT Eamplar Problems) ( Q sin for all in R 0 sin ) A ( Q cos + sin, for all in R) Also cos + sin 4 sin + sin 4 sin + 4 A 4 Thus, A and A 4 A and 4 A 4 A 4 Q sin 0 for all in R EXERCISE Very short answer type questions ( to ) : Write the domain of the following trigonometric functions : (i) sin (ii) cos (iii) tan (iv) cot (v) sec (vi) cosec Write the range of the following trigonometric functions : (i) sin (ii) cos (iii) tan (iv) cot (v) sec (vi) cosec What is the domain of the function f defined by f () sin? 4 Find the range of the following functions : (i) f () cos (ii) f () + sin Which of the si trigonometric functions are positive for the angles 4 (i) (ii) 7? 6 In which quadrant does lie if (i) cos is positive and tan is negative (ii) both sin and cos are negative (iii) sin 4 and cos (iv) sin and cos? 7 Find the values of the the following : (i) tan 4 (ii) sin (NCERT) (iii) sec 8 Find the values of the following : (i) cot 7 4 (ii) sin 7 (iii) cosec 9 Find the values of the following : (i) sin 76 (NCERT) (ii) tan 9 (iii) cos ( 070 )

23 TRIGNMETRIC FUNCTINS 0 If sin and lies in the second quadrant, find the value of cos If cos If tan 4 and lies in the third quadrant, find the value of sin and lies in the fourth quadrant, find the value of cos If cot and lies in the third quadrant, find the value of sin 4 Find the other five trigonometric functions if (i) cos and lies in the third quadrant (NCERT) (ii) cos and lies in the third quadrant (NCERT) (iii) cot 4 and lies in the third quadrant (NCERT) (iv) cot and lies in the second quadrant (NCERT) (v) tan 4 and does not lie in the first quadrant (vi) cosec and does not lie in the third quadrant If sin and lies in the second quadrant, show that sec + tan 6 If sin sec and lies in the second quadrant, find sin and sec 7 If sin : cos :: :, find sin, cos 8 If cos and < < cosec + cot, find the other t-ratios and hence evaluate sec tan 9 If tan 4, find the value of 9 sec 4 cot 0 If sec and + tan + cosec < <, find the value of + cot cosec If sec + tan, find the value of sec, tan, cos and sin In which quadrant does lie? If cosec cot, find cos In which quadrant does lie? Show that (i) sin 6 cos 0 + sin 4 cos 4 + sin cos (ii) sin 6 + cos tan 4 (iii) 4 sin sin + cos tan + cosec sec Evaluate sec 6 tan + sin 4 cosec 4 + cos 6 cot

24 TRIGNMETRIC FUNCTINS 99 EXERCISE ANSWERS (i) Third quadrant (ii) First quadrant (iii) (i) First quadrant (ii) Third or fourth quadrant (iii) Third quadrant (i) 4 (i) 7 6 (ii) 7 4 (ii) 6 9 (iii) 9 6 (iii) 40 4 (i) 00 (ii) 8 (iii) 864 (iv) 4 6 (i) 4 8 (ii) (iii) (i) (ii) 9 8 cm 0 cm 0 ; Indian railways is an important means of transportation for both human beings and goods Various goods such as coal, iron ore, heavy machinery etc are transported through railways in India Railways are playing a major role in the progress of the country : 44 cm 4 68 cm (i) 7 6 (ii) (iii) cm , 7 9 EXERCISE 4,, radians km 9 9 (i) R (ii) R (iii) { : R, (n + ), n I } (iv) { : R, n, n I} (v) { : R, (n + ), n I } (vi) { : R, n, n I} (i) [, ] (ii) [, ] (iii) R (iv) R (v) (, ] U [, ) (vi) (, ] U [, ) R 4 (i) [, ] (ii) [, 7] (i) tan, cot (ii) cos, sec 6 (i) fourth (ii) third (iii) second (iv) not possible as we must have sin + cos 7 (i) (ii) (iii) 8 (i) (ii) (iii) 9 (i) (ii) (iii) (i) sin, tan, cot, sec, cosec (ii) sin 4, tan 4, cot 4, sec, cosec 4 (iii) sin 4, cos, tan 4, sec, cosec 4 (iv) sin, cos, tan, sec, cosec

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