Question

If A, B and C are the angles of a triangle ABC, then `|(sin2"A", sin"C", sin"B"),(sin"C", sin2"B", sin"A"),(sin"B", sin"A", sin2"C")|` = ______.

Options

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Answer 1

If A, B and C are the angles of a triangle ABC, then `|(sin2"A", sin"C", sin"B"),(sin"C", sin2"B", sin"A"),(sin"B", sin"A", sin2"C")|` = 0.

Explanation:

Since, `"a"/sin"A" =  "b"/sin"B" = "c"/sin"C"` = 2R

and cosA = `("b"^2 + "c"^2 - "a"^2)/(2"bc") = "a"(("b"^2 + "c"^2 - "a"^2))/(2"abc")`

∴ sin2A = 2sinA cosA = `2."a"/(2"R").(("b"^2 + "c"^2 - "a"^2))/(2"bc")`

Hence the elements of R₁ in ∆ are `("a"("b"^2 + "c"^2 - "a"^2))/(2"Rbc"), "c"/(2"R"), "b"/(2"R")`

or `1/(2"Rbc"){"a"("b"^2 + "c"^2 - "a"^2)"bc"^2"b"^2"c"}`

∴  Δ = `1/((2"Rbc")(2"Rca")(2"Rab"))`

`|("a"("b"^2 + "c"^2 - "a"^2), "bc"^2, "b"^2"c"),("c"^2"a", "b"("c"^2 + "a"^2 - "b"^2), "a"^2"c"),("b"^2"a", "ba"^2, "c"("a"^2 + "b"^2 - "c"^2))|`

Take a, b, c common from C₁, C₂ and C₃ respectively.

∴  Δ =  `1/(8"R"^3"abc")`

`|("b"^2 + "c"^2 - "a"^2, "c"^2, "b"^2),("c"^2, "c"^2 + "a"^2 - "b"^2, "a"^2),("b"^2, "a"^2, "a"^2 + "b"^2 - "c"^2)|`

Now apply R₁ – R₂ and R₂ – R₃ and take (b² – a²) and (c² – b²) common from R₁ and R₂.

∴  Δ = `(("b"^2 - "a"^2)("c"^2 - "b"^2))/(8"R"^3"abc")|(1, 1, 1),(1, 1, 1),("b"^2, "a"^2, "a"^2 + "b"^2 - "c"^2)|`

∴  Δ = 0 (as two rows are Identical)