Mathematics

Ques: If   then find the value of   ? Graphical Approach: Let us take 'x' in place of  . Graph of  : it is a periodic function, because cot x is periodic and the period is  . So, we have to draw graph for a period   and repeat for entire values of x. Range of   =  so Range of   =  We know that,   , if  since,  , if   or  so, first draw the line   for   and repeat it for entire x. Similarly, draw the graph of   , if   or  and range of  Draw both the graph together to get the difference, Clearly, in interval   the difference is pi, both the graph are represented by lines in this interval which are parallel. Thus, throughout the interval difference will be the same i.e. pi . i.e.    ( since graph of arccotx is above arctanx, therefore in difference put minus sign) and second observation, in interval,  Traditional approach, since  and,  therefore first we convert the interval to (0,pi/2) If  and   is acute a

Ques: If   then find the value of x, where [.] denotes the greatest integer function. Sol. Since [.] gives the integer value and both the functions i.e.   and    are always greater than '0'. Therefore, sum of    when BOTH are '0'. STEPS : We separately find the interval of x for both the expression i.e.   and     for which they are zero and then take the INTERSECTION of both the intervals. 1. First consider   , see the graph, Clearly, from the graph,  , if   or    ..................... (1) 2. Consider,  understand the below graph, Clearly from the graph,  , if   or   .................... (2) Now, we have to take the intersection of (1) and (2) i.e. to find the values of x for which both   and  are '0'. so,  But to take intersection of these two intervals we have to find where these numbers viz. cos1, cot1 and 1 lie on number line. since,   and we know that sin1 < 1 therefore, cot1 > cos1 we know that from above

The question is   and we have to find its domain we know that the cos inverse is defined between -1 and 1 therefore,  here denominator,   and   because  since,   we can multiply by it in above inequality, so, inequality becomes, there are two inequalities, and  take one by one, 1) when,  or,   or,  since sinx is always greater than -6 , therefore it is true for all x, so   or,  2) when  or,  we know that sin x is greater than 'zero' between 0 and pi. so,  since both inequalities (1) and (2) must be satisfied, the domain is intersection of both therefore,  answer is