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

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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.