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In a triangle, the three angles A, B, C satisfy tan A = 2 and tan B =…
Question
In a triangle, the three angles A, B, C satisfy tan A = 2 and tan B = 3. Then cot A + cot B + cot C can be written as m/n in lowest terms with positive integers m, n. Find m + n.
✓ Verified answer: 17checked by our engine — not a guess
Step-by-step solution
Since A + B + C = π, we have C = π − (A + B), so tan C = −tan(A + B) = −(tan A + tan B)/(1 − tan A tan B) = −(2 + 3)/(1 − 2·3) = −5/(−5) = 1.
(All of tan A, tan B, tan C are positive, consistent with an acute triangle.)
Then cot A + cot B + cot C = 1/2 + 1/3 + 1/1 = 3/6 + 2/6 + 6/6 = 11/6.
In lowest terms m/n = 11/6, so m + n = 11 + 6 = 17.
Final answer17
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