RELATIONS BETWEEN SIDES AND ANGLES OF A TRIANGLE

MODULE - IV 19 RELATIONS BETWEEN SIDES AND ANGLES OF A TRIANGLE In an earlier lesson, we have learnt about trigonometric functions of real numbers, relations between them, drawn the graphs of trigonometric functions, studied the characteristics from their graphs, studied about trigonometric functions of sum and difference of real numbers, and deduced trigonometric functions of multiple and sub-multiples of real numbers. We also studied about inverse trigonometric functions, some of their properties and solved problems based on their concepts. In this lesson, we shall try to establish some results which will give the relationship between sides and angles of a triangle and will help in finding unknown parts of a triangle. OBJECTIVES After studying this lesson, you will be able to : derive sine formula, cosine formula and projection formula apply these formulae to solve problems. EXPECTED BACKGROUND KNOWLEDGE Trigonometric and inverse trigonometric functions. Formulae for sum and difference of trigonometric functions of real numbers. Trigonometric functions of multiples and sub-multiples of real numbers. 19.1 SINE FORMULA In a ABC, the angles corresponding to the vertices A, B, and C are denoted by A, B, and C and the sides opposite to these vertices are denoted by a, b and c respectively. These angles and sides are called six elements of the triangle. MATHEMATICS 159

MODULE - IV Result 1 : Prove that in any triangle, the lengths of the sides are proportional to the sines of the angles opposite to the sides, i.e. Proof : In ABC, in Fig. 19.1 [(i), (ii) and (iii)], BC a, CA b and AB c and C is acute angle in (i), right angle in (ii) and obtuse angle in (iii). Draw AD prependicular to BC In ABC, AD sinb AB or (i) (ii) (iii) Fig. 19.1 (or BC produced, if need be) AD sinb c AD c sin B...(i) In ADC, AD sinc in Fig 19.1 (i) AC or, AD sinc b AD b sin C...(ii) If Fig. 19.1 (ii), or AD π 1 sin sinc AC AD b sin C AD and in Fig. 19.1 (iii), sin ( C ) sinc AC π or Thus, in all three figures, AD b sin C From (i) and (ii), we get AD sinc b or AD b sin C c sin B b sin C b c...(iii) sinb sinc 160 MATHEMATICS

Similarly, by drawing prependicular from C on AB, we can prove that From (iii) and (iv), we get a b...(iv) sina sinb...(a) (A) is called the sine-formula Note : (A) is sometimes written as...(a') The relations (A) and (A') help us in finding unknown angles and sides, when some others are given. Let us take some examples : MODULE - IV Example 19.1 Prove that ( ) A b c sin Solution : R.H.S. ( + ) B C A acos b + c sin, using sine-formula. We know that, k (say) a k sin A, b k sin B, c k sin C A k sinb + sinc sin R.H.S. ( ) Now B + C 90 A B+ C B C A ksin cos sin ( A+ B+ Cπ) B+ C A sin cos R.H.S. A B C A kcos cos sin B C k sina cos B C a cos L.H.S MATHEMATICS 161

MODULE - IV Example 19. Using sine formula, prove that Solution : We have A ( ) ( ) a cosc cosb b c cos k (say) a k sin A, b k sin B, c k sin C A R.H.S k( sinb sinc) cos B+ C B C A k cos sin cos A B C A B C A 4ksin sin cos asin cos B+ C B C asin sin acosc ( cosb) L.H.S. Example 19.3 In any triangle ABC, show that Solution : We have asina bsinb csin( A B) k (say) L.H.S. ksina sina ksinb sinb k sin A sin B ksin( A+ B) sin( A B) A B C sin A+ B sinc + π ( ) L.H.S. ksinc sin( A B) csin( A B) R.H.S. Example 19.4 In any triangle, show that a( bcosc ccosb) b c 16 Solution : We have, k (say) MATHEMATICS

L.H.S. ksina( ksinbcosc ksinccosb) ( ) k sina sin B C k sin( B+ C) sin( B C) sin A sin( B+ C) ( ) k sin B sin C k sin B k sin C b c R.H.S MODULE - IV CHECK YOUR PROGRESS 19.1 1. Using sine-formula, show that each of the following hold : (i) A B tan a b A+ B tan a+ b (ii) bcosb+ ccosc acos( B C) (iii) asin B C ( b c) cos A (iv) b+ c B + C B C tan cot b c (v) acosa+ bcosb + ccosc asinbsinc. In any triangle if a cosa b, prove that the tiangle is isosceles. cosb 19. COSINE FORMULA Result : In any triangle, prove that (i) b + c a cosa bc (ii) Proof : c + a b cosb (iii) ac a + b c cosc ab (i) (ii) (iii) Fig. 19. MATHEMATICS 163

MODULE - IV Three cases arise : (i) When C is acute (ii) When C is a right angle (iii) When C is obtuse Let us consider these one by one : Case (i) When C is acute AD sinc AC AD bsin C Also BD BC DC a bcosc DC b cosc From Fig. 19. (i) c ( bsinc) + ( a bcosc) b sin C + a + b cos C abcosc a + b abcosc a + b c cosc ab Case (ii) When C 90 c AD + BD b + a As C 90 cos C 0 c b + a ab cosc Case (iii) When b + a c cosc ab C is obtuse AD sin ( 180 C) sinc AC AD bsinc Also, BD BC + CD a bcos + ( 180 C) a bcosc c ( bsinc) + ( a bcosc) a + b abcosc 164 a + b c cosc ab MATHEMATICS

a + b c In all the three cases, cosc ab Similarly, it can be proved that c + a b b + c a cosb and cosa ac bc Let us take some examples to show its application. MODULE - IV Example 19.5 In any triangle ABC, show that Solution : We know that cosa cosb cosc a + b + c + + abc b + c a c + a b a + b c cosa, cosb, cosc bc ac ab L.H.S. b + c a c + a b a + b c + + abc abc abc 1 b + c a + c + a b + a + b c abc [ ] a + b + c R.H.S. abc Example 19.6 If A 60, show that in ABC (a+ b + c)(b + c a) 3bc Solution : b + c a cosa bc...(i) A 60 cosa cos60 1 (i) becomes 1 b + c a bc b + c a bc or b + c + bc a 3bc or ( b+ c) a 3bc or ( b+ c + a)( b + c a ) 3bc. MATHEMATICS 165

MODULE - IV Example 19.7 If the sides of a triangle are 3 cm, 5 cm and 7 cm find the greatest angle of a triangle. Solution : Here a 3 cm, b 5 cm, c 7 cm We know that in a triangle, the angle opposite to the largest side is greatest C is the greatest angle. a + b c cosc ab 9+ 5 49 15 1 30 30 cosc 1 π C 3 The greatest angle of the triangle is 3π or 10. Example 19.8 In ABC, if A 60, prove that b + c a Solution : cosa bc b c + 1. c+ a a + b or 1 b + c a cos60 bc b + c a bc or b + c a + bc...(i) b c ab+ b + c + ac R.H.S. + c+ a a + b ( c+ a)( a + b) ab+ ac + a + bc ( c+ a) ( a + b) [Using (i)] a( a+ b) + c( a + b) ( a+ c) ( a + b) ( a+ c)( a+ b) 1 ( a+ c) ( a+ b) 166 MATHEMATICS

CHECK YOUR PROGRESS 19. MODULE - IV 1. In any triangle ABC, show that (i) b c c a a b sina+ sinb+ sinc 0 (ii) ( ) ( ) ( ) a b + c tanb b c + a tanc c a + b tana k a + b + c (iii) ( sina+ sinb + sinc) abc where k b c cota+ c a cot B + a b cotc 0 (iv) ( ) ( ) ( ). The sides of a triangle are a 9 cm, b 8 cm, c 4 cm. Show that 6 cos C 4 + 3 cos B. 19.3 PROJECTION FORMULA Result 3 : In ABC, if BC a, CA b and AB c, then prove that (i) a bcosc + ccosb (ii) b ccosa + acosc (iii) c acosb + bcosa Proof : (i) (ii) (iii) Fig. 19.3 As in previous result, three cases arise. We will discuss them one by one. (i) When In In C is acute : ADB, BD cosb c BD ccosb ADC, DC cosc b DC b cos C a BD + DC c cos B + b cos C a c cos B + b cos c MATHEMATICS 167

MODULE - IV (ii) When C 90 (iii) When In ADB, BC a BC AB cosb c AB ccosb+ 0 ccosb + bcos90 ( cos90 0) ccosb + bcosc C is obtuse BD cosb c BD c cos B CD In ADC, cos ( C) cosc b π CD bcosc In Fig.19.3 (iii), BC BD CD a ccosb ( bcosc) ccosb + bcosc 168 Thus in all cases,a bcosc+ ccosb Similarly, we can prove that b ccosa + acosc and c acosb + bcosa Let us take some examples, to show the application of these results. Example 19.9 In any triangle ABC, show that ( b+ c) cosa+ ( c + a) cosb + ( a + b) cosc a + b c+ Solution : L.H.S. bcosa+ ccosa+ ccosb+ acosb+ acosc + bcosc ( bcosa+ acosb) + ( ccosa+ acosc) + ( ccosb+ bcosc) c+ b+ a a + b + c R.H.S. Example 19.10 In any ABC, prove that cosa cosb 1 1 a b a b 1 sin A 1 sin B Solution : L.H.S. a b 1 sin A 1 sin B + a a b b 1 1 + 1 1 a b a b R. H. S. k k sina sinb k a b MATHEMATICS

Example 19.11 In ABC, if acosa bcosb, where a b prove that ABC is a right angled triangle. MODULE - IV Solution : acosa bcosb a b + c a b a + c b bc ac or a ( ) b ( ) b + c a a + c b or 4 4 a b + ac a ab + bc b or ( ) ( )( ) c a b a b a + b c a + b ABC is a right triangle. Example 19.1 If a, b 3, c 4, find cos A, cos B and cos C. Solution : and b + c a cosa bc 9+ 16 4 1 7 3 4 4 8 c + a b 16 + 4 9 11 cosb ac 4 16 a + b c 4+ 9 16 3 1 cosc ab 3 1 4 CHECK YOUR PROGRESS 19.3 1. If a 3, b 4 and c 5, find cos A, cos B and cos C.. The sides of a triangle are 7 cm, 4 3cm and 13cm. Find the smallest angle of the triangle. 3. If a : b : c 7 : 8 : 9, prove that cos A : cos B : cos C 14 : 11 : 6. 4. If the sides of a triangle are x + x + 1, x+ 1 and x 1. Show that the greatest angle of the triangle is 10. 5. In a triangle, b cos A a cos B, prove that the triangle is isosceles. 6. Deduce sine formula from the projection formula. MATHEMATICS 169

MODULE - IV It is possible to find out the unknown elements of a triangle, if the relevent elements are given by using Sine-formula : (i) Cosine foumulae : (ii) LET US SUM UP b + c a cosa bc c + a b cosb ac a + b c cosc ab Projection formulae : a bcosc + ccosb b ccosa + acosc c acosb + bcosa SUPPORTIVE WEB SITES http://www.wikipedia.org http://mathworld.wolfram.com TERMINAL EXERCISE In a triangle ABC, prove the following (1-10) : 1. asin( B C) + bsin( C A) + csin( A B) 0. acosa+ bcosb + ccosc asinbsinc 3. b c c a a b sina+ sinb + sinc 0 170 MATHEMATICS

4. c + a 1 + cosbcos( C A) b + c 1 + cosacos( B C) MODULE - IV 5. 6. c bcosa cosb b ccosa cosc a bcosc sinc c bcosa sin A 7. ( ) A B C a+ b+ c tan + tan c cot 8. A B a b C sin cos c 9. (i) bcosb+ ccosc acos( B C) (ii) acosa+ bcosb ccos( A B) 10. ( ) B ( ) B b c a cos + c + a sin 11. In a triangle, if b 5, c 6, 1. In any ABC, show that cosa b acosc cosb a bcosc A 1 tan, then show thata 41. MATHEMATICS 171

MODULE - IV ANSWERS CHECK YOUR PROGRESS 19.3 1. 4 cosa 5 3 cosb 5 cosc zero. The smallest angle of the triangle is 30. 17 MATHEMATICS