TRIGONOMETRIC R ATIOS & IDENTITIES

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1 TRIGONOMTRIC R ATIOS & IDNTITIS. INTRODUCTION TO TRIGONOMTRY : The word 'trigonometry' is derived from the Greek words 'trigon' and 'metron' and it means 'measuring the sides of a triangle'. The subject was originally developed to solve geometric problems involving triangles. It was studied by sea captains for navigation, surveyor to map out the new lands, by engineers and others. Currently, trigonometry is used 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 a musical tone and in many other areas. ( a ) Measurement of angles : There are three systems of measurement of angles. ( i ) Sexagesimal or nglish System : Here right angle 90 (degrees) 60' (minutes) ' 60" (seconds) (ii) Centesimal or rench System : Here right angle 00 g (grades) g 00' (minutes) ' 00" (seconds) ( i i i) Circular system : Here an angle is measured in radians. One radian corresponds to the angle subtended by an arc of length 'r ' at the centre of the circle of radius r. It is a constant quantity and does not depend upon the radius of the circle. ( b ) Relation between the three systems : D G R / ( c ) If is the angle subtended at the centre of a circle of radius 'r', by an arc of length '' then r. r Note that here, r are in the same units and is always in radians. Illustration : If the arcs of same length in two circles subtend angles of 60 and 75 at their centres. ind the ratio of their radii. Solution : Let r and r be the radii of the given circles and let their arcs of same length s subtend angles of 60 and 75 at their centres. Now, s r and r c 5 s r c and s 5 5 and r s r r 4r 5r r : r 5 : 4 A ns. Do yourself - : (i) xpress in the three systems of angular measurement, the magnitude of the angle of a regular decagon. (ii) The radius of a circle is 0 cm. ind the length of an arc of this circle if the length of the chord of the arc is 0 cm. c c NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory

2 . T-R ATIOS (or Trigonometric functions) : h p b p p In a right angle triangle sin ; cos ; tan ; h h b cos ec h ; sec h p b and cot b p 'p' is perpendicular ; 'b' is base and 'h' is hypotenuse. b Note : The quantity by which the cosine falls short of unity i.e. cos, is called the versed sine of and also by which the sine falls short of unity i.e. sinis called the coversed sine of.. BASIC TRIGONOMTRIC IDNTITIS : ( ) sin. cosec ( ) cos. sec sin cos ( ) tan. cot ( 4 ) tan & cot cos sin ( 5 ) sin + cos or sin cos or cos sin ( 6 ) sec tan or sec + tan or tan sec ( 7 ) sec + tan sec tan ( 8 ) cosec cot or cosec + cot or cot cosec NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory ( 9 ) cosec + cot cos ec cot ( 0 ) xpressing trigonometrical ratio in terms of each other : Illustration : Solution : Illustration : sin cos tan cot sec cosec sin sin cos cos sin cos tan sec tan cot sec cosec cot cosec tan cot sec cosec sin cos tan tan sec cos cot sin cosec cot sin cos cot cosec sin tan cos sec cot cosec sec tan sec cos cot sin cosec tan sec cosec cot cosec sin tan cos sec If sin sin, then prove that Given that sin sin cos cos cos cos cos 0 L.H.S. cos 6 (cos + ) sin ( + sin) (sin + sin ) 0 (sin 6 + cos 6 ) ( sin 4 + cos 4 ) + is equal to (A) 0 (B) (C) (D) none of these Solution : [(sin + cos ) sin cos ( sin + cos ) ] [ (sin + cos ) sin cos ] + [ sin cos ] [ sin cos ] + 6 sin cos + 6 sin cos + 0 Ans.(A)

3 Do yourself - : (i) (ii) 4 If cot, then find the value of sin, cos and cosec in first quadrant. If sin + cosec, then find the value of sin 8 + cosec 8 4. NW DINITION O T-RATIOS : By using rectangular coordinates the definitions of trigonometric functions can be extended to angles of any size in the following way (see diagram). A point P is taken with coordinates (x, y). The radius vector OP has length r and the angle is taken as the directed angle measured anticlockwise from the x- axis. The three main trigonometric functions are then defined in terms of r and the coordinates x and y. P(x, y) r y O x sin y/r, cos x/r tan y/x, (The other function are reciprocals of these) This can give negative values of the trigonometric functions. 5. SIGNS O TRIGONOMTRIC UNCTIONS IN DIRNT QUADRANTS : II quadrant only sine & cosec +ve 90, / I quadrant All +ve 80, only tan & cot +ve only cos & sec +ve 0, 60, III quadrant IV quadrant 70, / 6. TRIGONOMTRIC UNCTIONS O ALLID ANGLS : ( a ) sin (n + ) sin, cos (n + ) cos, where n I ( b ) sin () sin cos ( ) cos sin(90 ) cos sin(90 + ) cos sin(80 ) sin sin(80 + ) sin sin(70 ) cos sin(70 + ) cos sin (60 ) sin sin (60 + ) sin cos(90 ) sin cos(90 + ) sin cos(80 ) cos cos(80 + ) cos cos(70 ) sin cos(70 + ) sin cos(60 ) cos cos(60 + ) cos 70 NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory

4 7. VALUS O T-RATIOS O SOM STANDARD ANGLS : Angles T-ratio 0 /6 /4 / / / sin 0 / / / 0 cos / / / 0 0 tan 0 / N.D. 0 N.D. cot N.D. / 0 N.D. 0 sec / N.D. N.D. cosec N.D. / N.D. N.D. Not Defined (a) sin n 0 ; cos n ( ) n ; tan n 0 where n I (b) sin(n+) ( )n ; cos(n+) 0 where n I Illustration 4 : If sin and tan then is equal to - (A) 0 (B) 50 (C) 0 (D) none of these Solution : Let us first find out lying between 0 and 60. Since sin 0 or 0 and tan 0 or 0 Hence, 0 or 7 6 is the value satisfying both. Ans. (C) NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory Do yourself - : (i) If cos and, then find the value of 4tan cosec. (ii) Prove that : (a) cos570 sin50 + sin( 0 ) cos( 90 ) 0 (b) 9 7 tan sin cosec 4 cos GRAPH O TRIGONOMTRIC UNCTIONS : (i) y sinx (ii) y cosx Y Y X' / / / X o X' / o / / Y' Y' 7 X

5 (iii) y tanx (iv) y cotx Y Y X' o X X' o X Y' Y' ( v ) y secx (vi ) y cosecx Y Y (-,) (0,) (,) Y Y X' 5 /,0 /,0 /,0 /,0 /,0 5 /,0 o X X',0,0 o X (, ) (, ) Y Y Y' Y' 9. DOMAINS, RANGS AND PRIODICITY O TRIGONOMTRIC UNCTIONS : T-Ratio D o m a i n R a n g e Per i od sin x R [,] cos x R [,] tan x R {(n+)/ ; ni} R cot x R {n : n I} R sec x R {(n+) / : n I} ( ] [,) cosec x R {n : n I} (, ] [,) 0. TRIGONOMTRIC R ATIOS O TH SUM & DIRNC O TWO ANGLS : (i) sin (A + B) sin A cos B + cos A sin B. (ii) sin (A B) sin A cos B cos A sin B. (iii) cos (A + B) cos A cos B sin A sin B (iv) cos (A B) cos A cos B + sin A sin B ( v ) tan A tan B tan A tan B tan (A + B) (vi ) tan (A B) tan A tan B tan A tan B cot B cot A (vii) cot (A + B) (viii) cot (A B) cot B cot A cot B cot A cot B cot A Some more results : (i) sin A sin B sin (A + B). sin(a B) cos B cos A. (ii) cos A sin B cos (A+B). cos (A B). Illustration 5 : Prove that cosec0 sec0 4. Solution : L.H.S. cos 0 sin 0 sin 0 cos 0 sin 0.cos 0 4 cos 0 sin 0 sin 0cos 0 sin(60 0 ) sin R.H.S. sin 40 sin (sin 60.cos0 cos 60.sin 0 ) sin 40 NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory

6 Illustration 6 : Prove that tan70 cot70 + cot40. Solution : L.H.S. tan 0 tan 50 tan 70 tan(0 50 ) tan 0 tan 50 or tan70 tan0 tan50 tan70 tan0 + tan50 or tan70 tan70 tan50 tan0 + tan0 + tan50 tan 50 + tan0 cot70 + cot40 R.H.S. Do yourself - 4 : (i) If sin A and 5 9 cos B, 0 A & B, then find the value of the following : 4 (a) sin(a + B) (b) sin(a B) (c) cos(a + B) (d) cos(a B) (ii) If x + y 45, then prove that : (a) ( + tanx)( + tany) (b) (cotx )(coty ). ORMULA TO TRANSORM TH PRODUCT INTO SUM OR DIRNC : (i) sin A cos B sin (A+ B) + sin (A B). (ii) cos A sin B sin (A + B) sin (A B). (iii) cos A cos B cos (A + B) + cos (A B) (iv) sin A sin B cos (A B) cos (A + B) Illustration 7 : If sina sinb, then prove that tan(a B) tan(a B) Solution : Given sina sinb NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory sin A sin B Applying componendo & dividendo, sin A sin B sin B sin A A B A B sin cos B A B A cos sin sin(a B) cos(a B) cos(a B) sin{ (A B)} sin(a B) cos(a B) cos(a B) sin(a B) tan(a B) tan(a B) 7 sin(a B) cos(a B) cos(a B) sin(a B) ( ) tan(a B) cot(a B)

7 . ORMULA TO TRANSORM SUM OR DIRNC INTO PRODUCT : (i) sin C + sin D sin (iii) cos C + cos D cos C D C D cos cos C D C D (ii) sin C sin D cos (iv) cos C cos D sin C D C D sin C D sin D C sin 5 sin sin Illustration 8 : cos 5 cos cos cos is equal to - (A) tan (B) cos (C) cot (D) none of these sin cos sin sin cos Solution : L.H.S. cos.cos cos cos cos cos cos sin cos cos cos cos Illustration 9 : Show that sin.sin48.sin54 /8 sin ( cos ) tan cos ( cos ) Ans. (A) Solution : L.H.S. cos 6 cos 60sin 54 cos 6 sin 54 sin 54 cos 6sin 54 sin 54 sin 90 sin8 sin (sin 54 sin8 ) sin8cos 6 Do yourself - 5 : (i) (ii) Simplify Prove that sin8 sin 6 cos 6 cos8 cos 6 4 cos8 4 cos8 sin 6cos 6 sin 7 R.H.S. 4 cos8 4 sin sin 75 sin5 cos 75 cos5 (a) (sina + sina)sina + (cosa cosa)cosa 0 (b) cos0 cos40 cos60 cos80 6 (c) sin 8 cos sin 6 cos tan cos cos sin sin 4 74 NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory

8 . TRIGONOMTRIC R ATIOS O SUM O MOR THAN TWO ANGLS : ( i ) sin (A+B+C) sinacosbcosc + sinbcosacosc + sinccosacosb sinasinbsinc sina cosb cosc sin A cosa cosb cosc [tana + tanb + tanc tana tanb tanc] (ii) cos (A+B+C) cosa cosb cosc sina sinb cosc sina cosb sinc cosa sinb sinc cos A sin A sin B cos C cos A cos B cos C [ tan A tan B tan B tan C tan C tan A ] (iii) tan A tan B tan C tan A tan B tan C S S tan (A + B+ C) tan A tan B tan B tanc tan C tan A S Do yourself - 6 : Prove the above identities 4. TRIGONOMTRIC R ATIOS O MULTIPL ANGLS : ( a ) Trigonometrical ratios of an angle in terms of the angle : (i) sin sin cos tan tan (ii) cos cos sin cos sin tan tan (iii) + cos cos (iv) cos sin cos sin ( v ) tan tan (vi ) tan sin cos tan Illustration 0 : Prove that : cos A tan(60 A ) tan(60 A ). cos A NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory Solution : R.H.S. tan(60 + A) tan(60 A) tan 60 tan A tan 60 tan A tan 60 tan A tan 60 tan A tan A tan A tan A tan A sin A tan A cos A cos A sin A tan A sin A cos A sin A cos A (cos A sin A ) cos A sin A (cos A sin A ) (sin A cos A ) 75 cos A cos A sin A sin A cos A sin A sin A cos A cos A L.H.S. cos A Do yourself - 7 : (i) Prove that : (a) sin tan (b) sin cos cot cos sin cos (c) sec 8 tan 8 sec 4 tan

9 ( b ) Trigonometrical ratios of an angle in terms of the angle : (i) sin sin 4sin. (ii) cos 4cos cos. tan tan (iii) tan tan Illustration : Prove that : tana + tan(60 + A) + tan(0 + A) tana Solution : L.H.S. tana + tan(60 + A) + tan(0 + A) tana + tan(60 + A) + tan{80 (60 A)} tana + tan(60 + A) tan(60 A) [ tan(80 ) tan] tan 60 tan A tan 60 tan A tan A tan 60 tan A tan 60 tan A tan A tan A tan A tan A tan A tan A tan A tan A tan A tan A tan A tan A ( tan A)( tan A) tan A 8 tan A tan A tan A tan A 8 tan A tan A 9 tan A tan A tan A tan A tan A tan A R.H.S. tan A Do yourself - 8 : (i) Prove that : (a) cot cot (60 ) cot (60 + ) cot (b) cos5 6cos 5 0 cos + 5 cos (c) sin 4 4sin cos 4cos sin 5. TRIGONOMTRIC R ATIOS O SUB MULTIPL ANGLS : Since the trigonometric relations are true for all values of angle, they will be true if instead of be substitute tan (i) sin sin cos tan (ii) cos cos sin cos sin (iii) + cos cos (iv) cos sin ( v ) cos sin tan sin cos 76 tan tan NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory

10 (vi ) tan tan tan cos (vii) sin ± (viii) cos cos (ix) cos tan cos (x) sin sin sin ( xi) cos sin sin (xii) tan tan tan Q P sin cos is ve sin cos is ve sin cos is ve sin cos is ve sin cos is ve O sin cos is ve sin cos is ve R sin cos is ve S Illustration : sin Solution : 67 + cos 67 is equal to (A) 4 (B) 4 sin 67 + cos 67 (C) 4 (D) 4 4 sin 5 (using cosa + sina sin A ) 4 Ans.(A) 4 NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory Do yourself - 9 : (i) ind the value of (a) sin (b) cos (c) tan TRIGONOMTRIC R ATIOS O SOM STANDARD ANGLS : (i) (iii) ( v ) (vii) 5 5 sin 8 sin cos 7 cos (ii) cos 6 cos sin 54 sin sin 7 sin cos8 cos (iv) sin 6 sin cos 54 cos sin 5 sin cos 75 cos (vi ) cos5 cos sin 75 sin 5 5 tan 5 tan co t 7 5 co t (viii) tan 75 tan cot5 cot tan.5 tan cot 67.5 cot (x) tan 67.5 tan cot.5 cot (ix) 77

11 Illustration : valuate sin78 sin66 sin4 + sin6. Solution : The expression (sin78 sin4 ) (sin66 sin6 ) cos(60 ) sin(8 ) cos6. sin0 5 5 sin8 cos6 4 4 Do yourself - 0 : (i) ind the value of (a) sin sin (b) 0 0 cos 48 sin 7. CONDITIONAL TRIGONOMTRIC IDNTITIS : If A + B + C 80, then (i) tan A + tan B + tan C tan A tan B tan C (ii) cot A cot B + cot B cot C + cot C cot A A B B C C A ( i i i) tan tan tan tan tan tan (iv) cot A cot B cot C cot A cot B cot C ( v ) sin A + sin B + sin C 4 sina sinb sinc (vi ) cos A + cos B + cos C 4 cosa cosb cosc (vii) sin A + sin B + sin C 4 cos A B C A B C cos cos (viii) cos A + cos B + cos C + 4 sin sin sin Illustration 4 : In any triangle ABC, sin A cos B cos C, then angle B is (A) / (B) / (C) /4 (D) /6 Solution : We have, sin A cos B cos C sin A cos B + cos C A A B C B C sin cos cos cos A A A B C sin cos cos cos A + B + C A A A B C sin cos sin cos A B C cos cos or A B C ; But A + B + C Therefore B B / Illustration 5 : If A + B + C, then cos A + cos B + cosc is equal to- Solution : (A) 4cosA cosb cosc (B) 4 sina sin B sinc (C) + cosa cosb cosc (D) 4 sina sinb sinc cos A + cos B + cos C cos (A + B ) cos (A B) + cos C cos C cos (A B) + cos C A + B + C sin C cos ( A B) + sin C sinc [ cos ( A B) + sin C ) ] sin C [ cos (A B) + sin A B 78 Ans.(A) sin C [ cos (A B) cos ( A +B ) ] 4 sin A sin B sin C Ans.(D) NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory

12 Do yourself - : (i) If ABCD is a cyclic quadrilateral, then find the value of sina + sinb sinc sind (ii) If A + B + C, then find the value of tana tanb + tanbtanc + tanc tana 8. MA XIMUM & MINIMUM VALUS O TRIGONOMTRIC XPRSSIONS : (i) acos + bsin will always lie in the interval [ a b, a b ] i.e. the maximum and minimum (ii) values are a b, a b respectively. Minimum value of a tan + b cot ab where a, b (iii) a b ab cos( ) < a cos (+) + b cos () < a b ab cos( ) where and are known angles. and + (constant) then (iv) If,, 0, (i) Maximum value of the expression cos cos, cos + cos, sin sin or sin + sin occurs when / (ii) Minimum value of sec + sec, tan + tan, cosec + cosec occurs when / ( v ) If A, B, C are the angles of a triangle then maximum value of sin A + sin B + sin C and sin A sin B sin C occurs when A B C 60 (vi ) In case a quadratic in sin & cos is given then the maximum or minimum values can be obtained by making perfect square. NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory Illustration 6 : Prove that : 4 5 cos cos 0, for all values of. Solution : We have, 5cos + cos 5cos + coscos sin sin cos sin Since, cos sin 7 cos sin cos cos cos cos cos cos 0 79 for all. for all. for all. Illustration 7 : ind the maximum value of + sin 4 + cos 4 - (A) (B) (C) (D) 4 Solution : We have + sin cos 4 (cos + sin ) + ( cos + sin ) +. cos 4 maximum value. 4 (cos + sin ) Ans. (D)

13 Do yourself - : (i) ind maximum and minimum value of 5cos + sin 6 for all real values of. (ii) ind the minimum value of cos + cos for all real values of. (iii) ind maximum and minimum value of cos 6 sin cos sin. 9. IMPORTANT RSULTS : (i) sin sin (60 ) sin (60 +) sin 4 (ii) cos. cos (60 ) cos (60 + ) cos 4 (iii) tan tan (60 ) tan (60 + ) tan (iv) cot cot (60 ) cot (60 + ) cot ( v ) (a) sin + sin (60 +) + sin (60 ) (b) cos + cos (60 +) + cos (60 ) (vi ) (a) If tan A + tan B + tan C tan A tan B tan C, then A + B + C n, n I (b) If tan A tan B + tan B tan C + tan C tan A, then A + B + C (n + ),n I (vii) cos cos cos 4... cos ( n ) sin( ) n sin (viii) (a) cota tana cota (b) cota + tana coseca (ix) sin + sin () + sin (+) +... sin ( n ) (x) cos + cos (+) + cos ( + ) +... cos( n ) n R S T sin R S T cos U V I K J I K J n I K J U V ni sin W K J I sin K J n n sin W sin Illustration 8 : Prove that tana + tana + 4tan4A + 8cot8A cot A. Solution : 8 cot 8A cota tana tana 4tan4A cota tana 4tan4A (using viii (a) in above results) 4 cot4a 4tan4A (using viii (a) in above results) 8 cot8a. Aliter Method : L.H.S. tana + tana + 4tan4A + tana + tana + 4 tan 4A 4 4 tan 4A 80 tan 4A tana + tana + 4cot4A tana + tana + tan A tana + tan A tan A tan A tan A tan A 8 tan 4A 4 tana + cota tan A tan A tan A tan 4A tan A tan A cot A R.H.S. NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory

14 Illustration 9 : valuate Solution : Sum r n cos r n ; n n r cos n r 4 (n ) (n ) cos cos... cos n n n sin n (n ) (n ) n.cos n n sin n. n sin (n ) Using,cos cos( ) cos( )... cos( (n ) ).cos sin (n ) sin.cos (n ) n (n ) n sin n NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory r n n cos r n A ns. Illustration 0 : Prove that : ( + sec)( + sec )( + sec )...( + sec n ) tan n.cot. Solution : L.H.S.... n cos cos cos cos n cos cos cos cos... n cos cos cos cos n cos. cos. cos... cos n cos.cos.cos...cos cos(cos)(cos)(cos )...(cos n `). n cos cos sin (sincos)(cos)(cos )...(cos n `). n cos cos sin (sincos)(cos )...(cos n `). n cos cos sin (sinn.cos n `). n cos cos sin. sinn. n cos tann.cotr.h.s. Do yourself - : 5 (i) valuate sin + sin sin... to n terms n n n (ii) If ( n + ), then find the value of n n cos cos cos... cos. Miscellaneous Illustration : Illustration : Prove that tan + tan + tan n tan n n cot n cot Solution : We know tan cot cot...(i) Putting,,,...in (i), we get tan (cot cot ) (tan ) (cot cot ) (tan ) (cot cot )... n (tan n ) n (cot n cot n ) 8

15 Adding, tan + tan + tan n tan n cot n cot n tan + tan + tan n tan n + n cot n cot Illustration : If A,B,C and D are angles of a quadrilateral and Solution : A B C D /. A B C D sin sin sin sin A B A B C D C D cos cos cos cos Since, A + B (C + D), the above equation becomes, A B C D sin sin sin sin, prove that 4 A B A B C D A B cos cos cos cos A B A B A B C D A B C D cos cos cos cos cos cos 0 A B This is a quadratic equation in cos which has real roots. A B C D A B C D cos cos 4 cos. cos 0 A B C D cos cos 4 A B C D A B cos cos, Now both cos and A B C D cos & cos A B C D 0 A B, C D. Similarly A C, B D A B C D / C D cos : (i) 44, 60 g, : (i) 4 5,, 5 5 : (i) 8 4 : (i) (a) : (i) 9 : (i) (a) 4 5 (b) c 05 0 : (i) (a) : (i) 0 (ii) : (i) 7 & 7 (ii) 9 8 : (i) 0 (ii) ANSWRS OR DO YOURSL (ii) (ii) (c) (b) (b) cm (d) (iii) 4 0 & (c) NOD6 ()\Data\04\Kota\J-Advanced\SMP\Maths\Unit#0\ng\0. TRIGO RATIO & IDNTITIS\.Theory