prove that f(x)= 2 tan^-1(x) + sin^-1 { 2x/1+x^2} is a constant for all x>or= 1. find that constant?
Asked by pspratimasingh588
| 16th Aug, 2010,
10:38: PM
Expert Answer:
f(x) = 2tan-1(x) + sin-1 { 2x/1+x2}
Put x = tan θ
f(tan θ) = 2tan-1(tan θ) + sin-1 { 2tan θ/1+tan2 θ}
f(tan θ) = 2tan-1(tan θ) + sin-1 { 2tan θ/sec2 θ}
f(tan θ) = 2tan-1(tan θ) + sin-1 { 2sin θ cos θ}
f(tan θ) = 2θ + sin-1 { sin 2θ }
f(tan θ) = 2θ + 2θ
f(tan θ) = 4θ
Now x = tan θ > = 1
θ = π/4 for x = 1,
so the constant is 4θ = π.
Regards,
Team,
TopperLearning.
f(tan θ) = 2tan-1(tan θ) + sin-1 { 2tan θ/1+tan2 θ}
f(tan θ) = 2tan-1(tan θ) + sin-1 { 2tan θ/sec2 θ}
f(tan θ) = 2tan-1(tan θ) + sin-1 { 2sin θ cos θ}
f(tan θ) = 2θ + sin-1 { sin 2θ }
f(tan θ) = 2θ + 2θ
f(tan θ) = 4θ
Now x = tan θ > = 1
θ = π/4 for x = 1,
so the constant is 4θ = π.
Regards,
Team,
TopperLearning.
Answered by
| 17th Aug, 2010,
10:44: PM
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