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

Correct Answer is : π/4π/4π/4

Correct Answer is : π/4π/4π/4

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UPSC NDA I 2019 Mathematics Solved Paper

Question : 41 of 120

Marks: +1, -0

\tan \theta = \frac{1}{2}

\tan \varphi = \frac{1}{3},

then what is the value of (θ + φ) ?

Solution:

Concept:

The sum formula for tangent:

\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}

Explanation:

Given:

\tan \theta = \frac{1}{2}

and

\tan \varphi = \frac{1}{3}

Apply the formula for

\tan(\theta + \varphi)

\tan(\theta + \varphi) = \frac{\tan\theta + \tan\varphi}{1 - \tan\theta \tan\varphi}

Substitute the values:

\frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{2} \cdot \frac{1}{3}} = \frac{\frac{3+2}{6}}{1 - \frac{1}{6}} = \frac{\frac{5}{6}}{\frac{5}{6}} = 1

Thus,

\tan(\theta + \varphi) = 1

Since

1 = \tan\frac{\pi}{4}

, we get

\theta + \varphi = \frac{\pi}{4}

(principal values).

Answer:

\theta + \varphi = \frac{\pi}{4}

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