The equation sinx(sinx+cosx)=k has real solutions if and only if k is a real number such that_____________?

 

Asked by aahnik.mohanty | 22nd May, 2019, 10:09: PM

Expert Answer:

sinx(sinx+cosx)=k
sin2x + sinxcosx = k   .... (i)
 
begin mathsize 16px style We space know space that space
sin 2 straight x space equals space 2 sinxcosx
rightwards double arrow sinxcosx equals space fraction numerator sin 2 straight x over denominator 2 end fraction space space... space left parenthesis ii right parenthesis
cos 2 straight x space equals space 1 minus space 2 sin squared straight x space space
rightwards double arrow sin squared straight x space equals space 1 half space minus fraction numerator space cos 2 straight x over denominator 2 end fraction space space... space left parenthesis iii right parenthesis
Substituting space the space values space of space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis space in space the space equation space left parenthesis straight i right parenthesis comma space we space get
space 1 half space minus fraction numerator space cos 2 straight x over denominator 2 end fraction space plus space fraction numerator sin 2 straight x over denominator 2 end fraction space space equals space straight k
rightwards double arrow 1 minus space cos 2 straight x space plus space sin 2 straight x space equals space 2 straight k
rightwards double arrow sin 2 straight x space space minus cos 2 straight x space space equals space 2 straight k space minus space 1

Let space straight f left parenthesis straight x right parenthesis space equals space sinx space minus space cos space straight x space
space space space space space space space space space space space space equals space square root of 2 cross times space fraction numerator 1 over denominator square root of 2 end fraction open parentheses sinx space minus space cos space straight x close parentheses
space space space space space space space space space space space space equals space square root of 2 open parentheses fraction numerator 1 over denominator square root of 2 end fraction sinx space minus fraction numerator 1 over denominator square root of 2 end fraction cosx space close parentheses
space space space space space space space space space space space space equals space space sin space open parentheses straight x space minus space straight pi over 4 close parentheses
Range space of space the space sin space function space is space left square bracket negative 1 comma space 1 right square bracket
rightwards double arrow The space range space of space straight f left parenthesis straight x right parenthesis space will space be space open square brackets negative square root of 2 comma square root of 2 close square brackets. space space space space... space left parenthesis iv right parenthesis
We space have comma
sin 2 straight x space space minus cos 2 straight x space space equals space 2 straight k space minus space 1

rightwards double arrow negative square root of 2 less or equal than space 2 straight k space minus space 1 less or equal than square root of 2
rightwards double arrow open parentheses fraction numerator 1 minus square root of 2 over denominator 2 end fraction space close parentheses less or equal than straight k space less or equal than open parentheses fraction numerator 1 plus square root of 2 over denominator 2 end fraction space close parentheses space space space

Therefore space the space equation space sinx left parenthesis sinx plus cosx right parenthesis equals straight k space has space real space solutions space if space and space only space if space straight k space is space straight a space real space number space such space that space space open parentheses fraction numerator 1 minus square root of 2 over denominator 2 end fraction space close parentheses less or equal than straight k space less or equal than open parentheses fraction numerator 1 plus square root of 2 over denominator 2 end fraction space close parentheses. space space space space
space end style

Answered by Yasmeen Khan | 23rd May, 2019, 10:48: AM