In a ∆ABC, Given A = `sqrt(3) - 1` B = `sqrt(3) + 1` C = 60° Using cosine formula C2 = a2 + b2 – 2 ab cos C = `(sqrt(3) - 1)^2 + (sqrt(3) + 1)^2 - 2(sqrt(3) - 1) xx (sqrt(3) + 1) cos 60^circ` = `3 -2sqrt(3) + 1 + 3 + 2sqrt(3) + 1 - 2(3 - 1) xx 1/2` C2 = 8 – 2 = 6 ⇒ C = √6 Using since formula `"a"/sin"A" = "b"/sin"B" = "c"/sin"C"` `(sqrt(3) - 1)/sin"A" = (sqrt(3) + 1)/sin"B" = sqrt(6)/sin60^circ` `(sqrt(3) - 1)/sin"A" = sqrt(6)/sin 60^circ` `(sqrt(3) - 1)/sin"A" = sqrt(6)/(sqrt(3)/2)` ⇒ `(sqrt(3) - 1)/sin"A" = (2sqrt(3) * sqrt(2))/sqrt(3)` ⇒ `(sqrt(3) - 1)/sin"A" = 2sqrt(2)` ⇒ sin A = `(sqrt(3) - 1)/(2sqrt(2))` ......(1) sin(45° – 30°) = sin 45° . cos 30° – cos 45° sin 30° = `1/sqrt(2) * sqrt(3)/2 - 1/sqrt(2) * 1/2` sin 15° = `(sqrt(3) - 1)/(2sqrt(2))` ......(2) From equations (1) and (2), we have sin A = sin 15° ⇒ A = 15° In ∆ABC, We have A + B + C = 180° 15° + B + 60° = 180° B = 180° – 75° B = 105° ∴ The required sides and angles are C = `sqrt(6)`, A = 15°, B = 105° |