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JEE Main 2021 · Mathematics · Trigonometry and Inverse Trigonometric Functions · easy

In a triangle ABC, tan A=2 and tan B=3. Then angle C equals

Single correct option

In a triangle ABCABC, tan⁡A=2\tan A=2 and tan⁡B=3\tan B=3. Then angle CC equals
  1. A. π3\dfrac{\pi}{3}
  2. B. π6\dfrac{\pi}{6}
  3. C. 3π4\dfrac{3\pi}{4}
  4. D. π4\dfrac{\pi}{4}
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Answer: (D) π4\dfrac{\pi}{4}

Since A+B+C=πA+B+C=\pi, we have tan⁡C=−tan⁡(A+B)\tan C=-\tan(A+B). Using the addition formula, tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡Atan⁡B=2+31−6=−1.\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}=\frac{2+3}{1-6}=-1. So tan⁡C=1\tan C=1, giving C=π4C=\dfrac\pi4 (the other candidate 5π4\dfrac{5\pi}{4} is not an angle of a triangle). Consistency check: tan⁡A,tan⁡B>0\tan A,\tan B>0 means both are acute, so A+B>π2A+B>\dfrac\pi2 is expected, and C=π4C=\dfrac\pi4 then gives A+B=3π4A+B=\dfrac{3\pi}{4}. Also tan⁡A+tan⁡B+tan⁡C=6=tan⁡Atan⁡Btan⁡C\tan A+\tan B+\tan C=6=\tan A\tan B\tan C, as required.
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