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Question: If \(\tan A = - \frac{1}{2}\)and \(\tan B = - \frac{1}{3},\) then \(A + B =\)...

If tan⁡A=−12\tan A = - \frac{1}{2}and tan⁡B=−13,\tan B = - \frac{1}{3}, then A+B=A + B =

A

π4\frac{\pi}{4}

B

3π4\frac{3\pi}{4}

C

5π4\frac{5\pi}{4}

D

None of these

Answer

3π4\frac{3\pi}{4}

Explanation

Solution

We have tan⁡A=−12\tan A = - \frac{1}{2} and tan⁡B=−13\tan B = - \frac{1}{3}

Now, tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B=−12−131−12.13=−1\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A\tan B} = \frac{- \frac{1}{2} - \frac{1}{3}}{1 - \frac{1}{2}.\frac{1}{3}} = - 1

⇒tan⁡(A+B)=tan⁡3π4.\Rightarrow \tan(A + B) = \tan\frac{3\pi}{4}. Hence, A+B=3π4.A + B = \frac{3\pi}{4}.