Question

How many positive numbers \[x\] satisfy the equation \[\cos \left( {97x} \right) = x\] ?
A) \[1\]
B) \[15\]
C) \[31\]
D) \[49\]

Answer 1

Hint: In the above question, we are given a trigonometric equation \[\cos \left( {97x} \right) = x\] . We have to find the number of positive values for which the given equation \[\cos \left( {97x} \right) = x\] is satisfied. In order to approach the solution, first we have to check the period of \[\cos \left( {97x} \right)\] . As we know that the period of the standard trigonometric function of cosine, i.e. \[\cos x\] is \[2\pi \] , hence the period of \[\cos \left( {97x} \right)\] will be given by \[\dfrac{{2\pi }}{{97}}\].

Complete step by step answer:
The trigonometric function is \[\cos \left( {97x} \right) = x\] .
We have to find how many positive values of \[x\] satisfy the above given equation.
Since the period of cosines function \[\cos x\] is \[2\pi \] .
Hence, the period of \[\cos \left( {97x} \right)\] is \[\dfrac{{2\pi }}{{97}}\] .
The range of cosine function, \[\cos x\] is \[\left[ { - 1,1} \right]\] .
The range of cosine function, \[\cos x\] for only positive values is \[\left[ {0,1} \right]\] .
Now, since the period of \[\cos \left( {97x} \right)\] is \[\dfrac{{2\pi }}{{97}}\] ,
Therefore in between the interval \[\left[ {0,1} \right]\] , the functions repeats itself, i.e. makes oscillations for about the number of times equal to
\[ \Rightarrow \dfrac{1}{{\dfrac{{2\pi }}{{97}}}}\]
That is,
\[ \Rightarrow \dfrac{{97}}{2} \times \dfrac{7}{{22}}\]
Hence,
\[ \Rightarrow 15.438\]
These are \[15\]and almost a half times.
Also in each period, the two functions, \[\cos \left( {97x} \right)\] and \[x\] meet twice.
Therefore,
\[ \Rightarrow 15.438 \times 2\]
That gives,
\[ \Rightarrow 30.876\]
Hence, there is almost one more cycle completed which might have one more value of \[x\] .
Therefore, there are total possible positive values of \[x\] equal to the number of times,
\[ \Rightarrow 30 + 1\]
That is,
\[ \Rightarrow 31\]
That is the required number of positive values of \[x\] .
Therefore, there are \[31\] positive numbers \[x\] that satisfy the equation \[\cos \left( {97x} \right) = x\].

Hence, the correct option is (C).

Note:
We can also confirm our answer by the graphical method.
Let us suppose two functions \[x\] and \[\cos \left( {97x} \right)\] .
Now after plotting their graphs we can see that there are \[31\]intersections in the interval \[\left[ {0,1} \right]\] of both the functions.
The graph is given below.