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Chapter 5: Problem 11

If \(0\pi / 3\) (c) \(A(\operatorname{cosec} A)<\pi / 6\) (d) \(\sin (A) / A>\pi / 6\)

Short Answer

Expert verified
The correct solution is \(A(\operatorname{cosec} A)<\pi / 3\).

Step by step solution

01

Identify and Compare

Since \(A\) is positive and less than \(\pi / 6\), it means \(A\) is a small positive number. Thus, \(A(\operatorname{cosec} A)\) is very close to 1 (since \(\operatorname{cosec} A = 1/\sin A\), and \(\sin A \approx A\) for small \(A\)). Since \(1<\pi / 3\) and \(1<\pi / 6\), option (a) and option (c) are both feasible at this stage.
02

Check the Option (b) and (d)

For the other two options, we need to see if the ratio of \(\sin A / A\) is greater than \(\pi / 6\) and \(\pi / 3\). Since \(\sin A \leq 1\), for \(A<\pi / 6\), \(\sin A / A < 1 / A\) which in turn is definitely less than \(\pi / 3\) and \(\pi / 6\). Hence, these options are not feasible.
03

Decide Between Option (a) and (c)

Now, we need to determine whether \(A(\operatorname{cosec} A)\) is either less than \(\pi / 6\) or \(\pi / 3\). As mentioned before, \(A(\operatorname{cosec} A)\) is very close to 1. The correct option should be the one with the value greater than 1. Looking at options (a) and (c), we can see that \(\pi/3\) is greater than \(\pi/6\). Therefore, option (a) \(A(\operatorname{cosec} A)<\pi / 3\) is the correct option.

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