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JEE Main 2019 · 10 Jan · Shift II

JEE Main 2019 Mathematics question on Trigonometry

With the usual notation, in △ABC\triangle ABC, if ∠A+∠B=120°\angle A + \angle B = 120°, a=3+1a = \sqrt{3} + 1 and b=3−1b = \sqrt{3} - 1, then the ratio ∠A:∠B\angle A : \angle B is
  1. A.
    7:17:1
  2. B.
    3:13:1
  3. C.
    9:79:7
  4. D.
    5:35:3
Show answer and solution

Correct option: A

Solution

Using the tangent rule: tan((A-B)/2) = ((a-b)/(a+b)) · cot(C/2).

Since A+B = 120°, we have C = 60°, so cot(C/2) = cot(30°) = √3.

a-b = (√3+1)-(√3-1) = 2, and a+b = 2√3.

So tan((A-B)/2) = (2/2√3)·√3 = (1/√3)·√3 = 1, giving (A-B)/2 = 45°, so A-B = 90°.

Combined with A+B = 120°: A = 105°, B = 15°, giving A:B = 105:15 = 7:1.

[JEE Main 2019 · 10 Jan · Shift II]

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