Q40 / 71 · NTA Abhyas (JEE Main) 2020 · Mathematics · Inverse Trigonometric Functions

In a ΔABC, if ∠ A = ∠ B = 1 2 sin - 1 6 + 1 2 3 + sin - 1 1 3  and length of the side opposite to  ∠ C is   c = 6 ⋅ 3 1 4 , then the area of ΔABC is

Correct answer: 27

Solution

2 ∠ B = sin - 1 6 + 1 2 3 + sin - 1 1 3

       = tan - 1 6 + 1 3 - 2 + tan - 1 1 2

      = tan - 1 6 + 1 3 - 2 + 1 2 1 - 6 + 1 3 - 2 · 1 2

      = tan - 1 6 + 1 2 + 3 - 2 2 3 - 2 - 6 + 1

      = tan - 1 1 2 + 2 + 3 - 2 6 - 2 - 6 - 1

      = tan - 1 2 3 + 3 - 3

      = tan - 1 3 3 - 3

     = tan - 1 - 3

       = 2 π 3

According to question

∠ A = ∠ B = 1 2 sin - 1 6 + 1 2 3 + sin - 1 1 3

⇒ ∠ A = ∠ B = 1 2 2 π 3

⇒ ∠ A = ∠ B = π 3

∴ ∠ C = π - ∠ A - ∠ B

         = π - 2 π 3

        = π 3

Since Δ ABC  is an equilateral Δ

∴  ar  Δ ABC = 3 4 a 2

                        = 3 4 c 2

                         = 3 4 · 6⋅ 3 1 / 4 2

                       = 3 4 3 ⋅ 6 × 6

                       = 3 × 3 × 3

                       = 2 7

Hence, the correct answer is 27.

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