Well, got this question from R.D. Sharma,Class XI.(Maximum value and minimum value)
Ex.7.2
Q.4 Prove that (2square root of 3+3)sintheta+2square root of 3costhetalies between -(2square root of 3+square root of 15) and (2square root of 3+square root of 15).
 
 

Asked by Akshat Bharadwaj | 2nd Jun, 2014, 01:19: PM

Expert Answer:

A s s u m e comma space 2 square root of 3 plus 3 space equals space a space comma space space 2 square root of 3 equals b  s o space g i v e n space s t a t e m e n t space c a n space b e space w r i t t e n space a s space a space sin space theta space plus thin space b space cos space theta M u l t i p l y space a n d space d i v i d e space t h e space s t a t e m e n t space b y space square root of a squared plus b squared end root comma space w e space g e t  equals square root of a squared plus b squared end root open square brackets fraction numerator a space sin space theta space plus thin space b space cos space theta over denominator square root of a squared plus b squared end root end fraction close square brackets equals space square root of a squared plus b squared end root open square brackets fraction numerator a over denominator square root of a squared plus b squared end root end fraction S i n space theta space plus thin space fraction numerator b over denominator square root of a squared plus b squared end root end fraction C o s space theta close square brackets  F o r space a n y space a n g l e space alpha comma space i f space T a n space alpha space equals space a over b comma space t h e n space S i n space alpha equals fraction numerator a over denominator square root of a squared plus b squared end root end fraction comma space C o s space alpha space equals space fraction numerator b over denominator square root of a squared plus b squared end root end fraction S u b s t i t u t i n g space a b o v e space v a l u e s space i n space t h e space s t a t e m e n t comma space w e space g e t.  equals space square root of a squared plus b squared end root open square brackets S i n space alpha space sin space theta space plus C o s space alpha space cos space theta thin space close square brackets  w e space k n o w space t h a t comma space cos space A space cos space B space plus thin space S i n space A space S I n space B space equals space C o s space left parenthesis A space minus space B right parenthesis  T h e r e f o r e space a b o v e space s t a t e m e n t space b e c o m e s comma space square root of a squared plus b squared end root space open square brackets C o s space left parenthesis theta space minus alpha right parenthesis close square brackets  W e space k n o w space t h a t comma space m a x i m u m space a n d space m i n u m space v a l u e s space o f space C o s space f u n c t i o n space i s space plus 1 space a n d space minus 1 space r e s p e c t i v e l y. space S o space m a x i m u m space a n d space m i n i m u m space v a l u e space o f space a b o v e space s t a t e m e n t space b e c o m e s space square root of a squared plus b squared end root space comma space minus square root of a squared plus b squared end root  S u b s t i t u t e space v a l u e s space o f space a space a n d space b space i n space t h i s comma space w e space g e t space t h e space r e q u i r e d space a n s w e r space w h i c h space l i e s space b e t w e e n space t h e space g i v e n space r a n g l e. space  F o r space a space b e t t e r space u n d e r t s a n d i n g comma space c h e c k space t h e space a p p r o x i m a t e space v a l u e space o f space r a n g e space i n space c a l c u a l t o r space a n d space v e r i f y space w i t h space t h e space a b o v e space s o l u t i o n

Answered by Dharma Teja | 4th Jun, 2014, 03:07: PM