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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

If -(π)/(2) α_1 α_2 α_3 (π)/(2) , then number of values of θ...

If −π2<α1<α2<α3<π2-\frac{\pi}{2} \lt \alpha_1 \lt \alpha_2 \lt \alpha_3 \lt \frac{\pi}{2}, then number of values of θ∈(−π2,π2)\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) satisfying \left(\tan \theta-\tan \alpha_1\right)(tan⁡θ−tan⁡α2)(tan⁡θ−tan⁡α3)−(∑(tan⁡θ−tan⁡αi))\left(\tan \theta-\tan \alpha_2\right)\left(\tan \theta-\tan \alpha_3\right)-\left(\sum\left(\tan \theta-\tan \alpha_i\right)\right) is ______ .

Correct answer

3

Step-by-step solution

Let f(θ)=Πf(\theta)=\Pi\left(\tan \theta-\tan \alpha_i\right)-(∑(tan⁡θ−tan⁡αi))\left(\sum\left(\tan \theta-\tan \alpha_i\right)\right) We have f(θ)θ→−π/2→−∞f(\theta)_{\theta \rightarrow-\pi / 2} \rightarrow-\infty Again f(α1)>0,f(α3)<0\mathrm{f}\left(\alpha_1\right)\gt0, \mathrm{f}\left(\alpha_3\right) \lt 0 and f(θ)θ→π/2→∞\mathrm{f}(\theta)_{\theta \rightarrow \pi / 2} \rightarrow \infty ⇒\Rightarrow There will be 3 such values of ' θ\theta '
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