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using the principal values, express the following as a single angle: . 3 tan⁻¹(1/2) + 2 tan⁻¹(1/5) + sin⁻¹{142/65√(5)}

using the principal values, express the following as a single angle: .                                     3 tan⁻¹(1/2) + 2 tan⁻¹(1/5) + sin⁻¹{142/65√(5)}

Grade:12

1 Answers

Samyak Jain
333 Points
4 years ago
tan–1(1/2) + 2 tan–1(1/5) + sin–1(142/65\sqrt{5})
Use sin–1(x) = tan–1(x / \sqrt{1-x^2})
sin–1(142/65\sqrt{5}) = tan–1(142 / \sqrt{(65\sqrt{5})^2 - 142^2}) = tan–1(142/\sqrt{21125 - 20164})
                             = tan–1(142/\sqrt{961})  =  tan–1(142/31)
tan–1(1/2) + 2 tan–1(1/5) + sin–1(142/65\sqrt{5}) = tan–1(1/2) + tan–1(1/5) + tan–1(1/5) + tan–1(142/31)
 =  tan–1((1/2 + 1/5)/(1 – 1/2 . 1/5)) + tan–1(1/5) + tan–1(142/31)
 =  tan–1(7/9) + tan–1(1/5) + tan–1(142/31)  =  tan–1((7/9 + 1/5)/(1 – 7/9 . 1/5)) + tan–1(142/31)
 =  tan–1(22/19) + tan–1(142/31)
 = tan–1 ((22/19 + 142/31) / (1 – 22/19 .142/31)) + \pi   =   tan–1(–4/3) + \pi
 = \pi – tan–1(4/3)  =  180\degree – 53\degree  =  127\degree
[\pi is added above because tan–1(22/19) + tan–1(142/31) is positive while tan–1(–4/3) is negative.]

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