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Write the value of tan ( \(\frac{\pi}{8}\)): |
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Answer» We have tan (\(\frac{\pi}{8}\)) = tan ( \(\frac{1}{2},\frac{\pi}{4}\) ) Using the half angle formulae, \(tan(\frac{\alpha}{8})=\frac{sin\alpha}{1+cos\alpha}\) \(tan(\frac{\pi}{8})=\frac{sin(\frac{\pi}{4})}{1+cos(\frac{\pi}{4})}\) \(=\frac{\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}}=\frac{1}{1+\sqrt{2}}\) Given, \(secβ=\frac{-5}{3},\frac{\pi}{2}<β<\pi\) ∴ \(tan^2β=1-sec^2β\) \(=1-\frac{25}{9}=\frac{-16}{9}\) ⇒ \(tanβ=\frac{-4}{3}\) (∵ \(\frac{p}{2}<β<p\)) And \(cot\alpha=\frac{1}{2},\) \(\frac{\pi}{2}<\alpha<\frac{3\pi}{2}\) |
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