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Question: The value of \(5\cos\theta + 3\cos(\theta + \frac{\pi}{3}) + 3\) lies between...

The value of 5cos⁡θ+3cos⁡(θ+π3)+35\cos\theta + 3\cos(\theta + \frac{\pi}{3}) + 3 lies between

A

−4- 4and 4

B

−4- 4 and 6

C

−4- 4and 8

D

−4- 4 and 10

Answer

−4- 4 and 10

Explanation

Solution

5cos⁡θ+3cos⁡(θ+π3)+35\cos\theta + 3\cos(\theta + \frac{\pi}{3}) + 3

=5cos⁡θ+3[cos⁡θcos⁡π3−sin⁡θ.sin⁡π3]+35\cos\theta + 3\lbrack\cos\theta\cos\frac{\pi}{3} - \sin\theta.\sin\frac{\pi}{3}\rbrack + 3

=[5cos⁡θ+32cos⁡θ−332sin⁡θ]+3\lbrack 5\cos\theta + \frac{3}{2}\cos\theta - \frac{3\sqrt{3}}{2}\sin\theta\rbrack + 3

= (132cos⁡θ−332sin⁡θ)+3\left( \frac{13}{2}\cos\theta - \frac{3\sqrt{3}}{2}\sin\theta \right) + 3

∵\because −(132)2+(332)2≤(132cos⁡θ−332sin⁡θ)≤(132)2+(332)2- \sqrt{\left( \frac{13}{2} \right)^{2} + \left( \frac{3\sqrt{3}}{2} \right)^{2}} \leq \left( \frac{13}{2}\cos\theta - \frac{3\sqrt{3}}{2}\sin\theta \right) \leq \sqrt{\left( \frac{13}{2} \right)^{2} + \left( \frac{3\sqrt{3}}{2} \right)^{2}}

−7≤(132cos⁡θ−332sin⁡θ)≤+7∴- 7 \leq \left( \frac{13}{2}\cos\theta - \frac{3\sqrt{3}}{2}\sin\theta \right) \leq + 7\therefore −7+3≤(132cos⁡θ−332sin⁡θ)+3≤7+3- 7 + 3 \leq \left( \frac{13}{2}\cos\theta - \frac{3\sqrt{3}}{2}\sin\theta \right) + 3 \leq 7 + 3

⇒ −4≤(132cos⁡θ−332sin⁡θ)+3≤10- 4 \leq \left( \frac{13}{2}\cos\theta - \frac{3\sqrt{3}}{2}\sin\theta \right) + 3 \leq 10So, the value lies

between – 4 and 10.