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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2019 · Mathematics · Trigonometric Ratios & Identities

If θ=(1)/(2) and =(1)/(3) , then what is the value of (θ+ ) ?

If tan⁡θ=12\tan \theta=\frac{1}{2} and tan⁡φ=13\tan \varphi=\frac{1}{3}, then what is the value of (θ+φ)(\theta+\varphi) ?

Options

  1. A0
  2. Bπ6\frac{\pi}{6}
  3. Cπ4\frac{\pi}{4}
  4. Dπ2\frac{\pi}{2}

Correct answer

C. π4\frac{\pi}{4}

Step-by-step solution

tan⁡θ=12\tan \theta=\frac{1}{2}tan⁡ϕ=13tan⁡(θ+ϕ)=tan⁡θ+tan⁡ϕ1−tan⁡θtan⁡ϕ=12+131−(12)(13)=5656=1∴θ+ϕ=tan⁡−1(1)=π4\begin{aligned} & \tan \phi=\frac{1}{3} \\ & \tan (\theta+\phi)=\frac{\tan \theta+\tan \phi}{1-\tan \theta \tan \phi} \\ & =\frac{\frac{1}{2}+\frac{1}{3}}{1-\left(\frac{1}{2}\right)\left(\frac{1}{3}\right)}=\frac{\frac{5}{6}}{\frac{5}{6}}=1 \\ & \therefore \theta+\phi=\tan ^{-1}(1)=\frac{\pi}{4}\end{aligned}$$
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