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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

What is the value of ((3 π)/(8))?

What is the value of $\tan \left(\frac{3 \pi}{8}\right)?

Options

  1. A2−1\sqrt{2}-1
  2. B2+1\sqrt{2}+1
  3. C1−21-\sqrt{2}
  4. D−(2+1)-(\sqrt{2}+1)

Correct answer

B. 2+1\sqrt{2}+1

Step-by-step solution

∵tan⁡\because \quad \tan\left(\frac{3 \pi}{4}\right)=\tan (π−π4)=−1\left(\pi-\frac{\pi}{4}\right)=-1 Let 3π4=θ⇒tan⁡θ=−1\frac{3 \pi}{4}=\theta \quad \Rightarrow \quad \tan \theta=-1 Now, tan⁡θ=2tan⁡θ/21−tan⁡2θ/2\tan \theta=\frac{2 \tan \theta / 2}{1-\tan ^2 \theta / 2} Let tan⁡θ/2=x\tan \theta / 2=x∴−1=2x1−x2⇒x2−2x−1=0⇒x=1±2∴tan⁡3π8=1+2(∵0<3π8<π2)\begin{aligned} & \therefore \quad-1=\frac{2 x}{1-x^2} \\ & \Rightarrow \quad x^2-2 x-1=0 \quad \Rightarrow \quad x=1 \pm \sqrt{2} \\ & \therefore \quad \tan \frac{3 \pi}{8}=1+\sqrt{2} \quad\left(\because 0 < \frac{3 \pi}{8} < \frac{\pi}{2}\right) \end{aligned}$$
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