Let $$S=\left[-\pi, \frac{\pi}{2}\right)-\left\{-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3 \pi}{4}, \frac{\pi}{4}\right\}$$. Then the number of elements in the set $$\mid A=\{\theta \in S: \tan \theta(1+\sqrt{5} \tan (2 \theta))=\sqrt{5}-\tan (2 \theta)\}$$ is __________.
Answer (integer)
5
Solution
<p>Let $\tan \alpha = \sqrt 5$</p>
<p>$\therefore$ $$\tan \theta = {{\tan \alpha - \tan 2\theta } \over {1 + \tan \alpha \tan 2\theta }}$$</p>
<p>$\therefore$ $\tan \theta = \tan (\alpha - 2\theta )$</p>
<p>$\alpha - 2\theta = n\pi + \theta$</p>
<p>$\Rightarrow 3\theta = \alpha - n\pi$</p>
<p>$$ \Rightarrow \theta = {\alpha \over 3} - {{n\pi } \over 3}\,\,\,\,\,\,\,\,\,;\,n \in Z$$</p>
<p>If $\theta \in [ - \pi ,\,\pi /2]$ then $n = 0,1,2,3,4$ are acceptable</p>
<p>$\therefore$ 5 solutions.</p>
About this question
Subject: Mathematics · Chapter: Trigonometric Functions · Topic: Trigonometric Ratios and Identities
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