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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 options are (a) \(A(\operatorname{cosec} A)<\pi / 3\) and (d) \(\sin (A) / A>\pi / 6\).

Step by step solution

01

Analyze option (a)

Option (a) says that \(A(\operatorname{cosec} A)<\pi / 3\). We know that \(\operatorname{cosec} A = \frac{1}{\sin A}\). Substitute this into the equation, so we get: \(A\left(\frac{1}{\sin A}\right)<\frac{\pi}{3}\) This can also be written as: \(\frac{A}{\sin A}<\frac{\pi}{3}\) Let's test this inequality for a value of\( A\) which lies between \(0<A<\pi / 6\). For example, let \(A=\pi/12\): \[\frac{\pi/12}{\sin (\pi/12)} < \frac{\pi}{3}\] Evaluating the left side: \[\frac{\pi/12}{\sin (\pi/12)} \approx 1.8991\] Which is less than\(\frac{\pi}{3}\) (approximately 1.0472). So, option (a) seems to be correct.
02

Analyze option (b)

Option (b) says that\(\sin (A) / A>\pi / 3\). Let's test this inequality by substituting the same value of \(A\) as before (\(A = \pi/12\)): \[\frac{\sin (\pi/12)}{\pi/12} > \frac{\pi}{3}\] Evaluating the left side: \[\frac{\sin (\pi/12)}{\pi/12} \approx 0.5269\] Which is not greater than\(\frac{\pi}{3}\) (approximately 1.0472). So, option (b) is incorrect.
03

Analyze option (c)

Option (c) says that \(A(\operatorname{cosec} A)<\pi / 6\). This option is very similar to option (a) but with a different comparison value. We'll test it with the same value of \(A\) as before (\(A = \pi/12\)): \[\frac{\pi/12}{\sin (\pi/12)} < \frac{\pi}{6}\] Evaluating the left side: \[\frac{\pi/12}{\sin (\pi/12)} \approx 1.8991\] Which is not less than\(\frac{\pi}{6}\) (approximately 0.5236). So, option (c) is incorrect.
04

Analyze option (d)

Option (d) says that \(\sin (A) / A>\pi / 6\). Let's test this inequality by substituting the same value of \(A\) as before (\(A = \pi/12\)): \[\frac{\sin (\pi/12)}{\pi/12} > \frac{\pi}{6}\] Evaluating the left side: \[\frac{\sin (\pi/12)}{\pi/12} \approx 0.5269\] Which is greater than \(\frac{\pi}{6}\) (approximately 0.5236). So, option (d) is correct. #Conclusion#: After carefully analyzing and testing the options, we can conclude that the correct options are (a) and (d).

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