SATHEE: Trigonometrical Equations 3 Question 19

Trigonometrical Equations 3 Question 19

20. For all $θ$ in $[0, π / 2]$ , show that $\cos (\sin θ) \geq \sin (\cos θ)$ .

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Solution:

We have, $\cos θ + \sin θ = \sqrt{2} \frac{1}{\sqrt{2}} \cos θ + \frac{1}{\sqrt{2}} \sin θ$

$= \sqrt{2} \sin \frac{π}{4} \cdot \cos θ + \cos \frac{π}{4} \cdot \sin θ$

$= \sqrt{2} \sin \frac{π}{4} + θ$

$\Rightarrow \cos θ + \sin θ \leq \sqrt{2} < \frac{π}{2}$

$\begin{aligned} as, \sqrt{2} & = 1.4141, \frac{π}{2} = 1.57 (approx) \\ \Rightarrow & \cos θ + \sin θ & < \frac{π}{2} \\ Since, & \cos θ & < \frac{π}{2} - \sin θ \\ \Rightarrow & \sin (\cos θ) & < \sin \frac{π}{2} - \sin θ \\ \Rightarrow & \sin (\cos θ) & < \cos (\sin θ) \\ \Rightarrow & \cos (\sin θ) & > \sin (\cos θ) \end{aligned}$

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Trigonometrical Equations 3 Question 19

20. For all θ in [0,π/2], show that cos⁡(sin⁡θ)≥sin⁡(cos⁡θ).

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20. For all $θ$ in $[0, π / 2]$ , show that $\cos (\sin θ) \geq \sin (\cos θ)$ .