What will be the nature of the equation (sinθ)/θ for 0 < θ < π/2 if θ increases continuously?

Category: QuestionsWhat will be the nature of the equation (sinθ)/θ for 0 < θ < π/2 if θ increases continuously?
Editor">Editor Staff asked 11 months ago

What will be the nature of the equation (sinθ)/θ for 0 < θ < π/2 if θ increases continuously?
 
(a) Decreases
 
(b) Increases
 
(c) Cannot be determined for 0 < θ < π/2
 
(d) A constant function
 
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This interesting question is from Calculus Application topic in portion Application of Calculus of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct choice is (a) Decreases
 
To elaborate: Let, f(θ) = (sinθ)/θ
 
Differentiating both sides of (1) with respect to θ we get,
 
f’(x) = (θcosθ – sinθ)/θ^2  ……….(1)
 
Further, assume that F(θ) = θcosθ – sinθ
 
Then, F’(x) = -θsinθ – cosθ + cosθ
 
= -θsinθ
 
Clearly, F’(x) < 0, when 0 < θ < π/2
 
Thus, F(θ) < F(0), when 0 < θ < π/2
 
But F(0) = 0*cos0 – sin0 = 0
 
Thus, F(θ) < 0, when 0 < θ < π/2
 
Therefore, from (1) it follows that,
 
f’(θ) < 0 in 0 < θ < π/2
 
Hence, f(θ) = (sinθ)/θ is a decreasing function for 0 < θ < π/2
 
i.e., for 0 < θ < π/2, f(θ) = (sinθ)/θ steadily decreases as θ continuously increases.