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Can you do this?
When defining
f(x)=Sin^(-1)1/√(1+x^2) (x>=0)
please proof there is constant C which belongs to this,
f(x)=C-Tan^(-1)x
2 Answers
- ?Lv 611 months ago
Does "Sin^(-1)1/√(1+x^2)" mean "arcsin(1/√(1+x^2))?
Let θ = arcsin(1/√(1+x^2)). So sinθ = 1/√(1+x^2).
x ≧ 0, so we can draw a right triangle △PQR.
............. ........... P
............. ...... * .. *
............. . * ....... *
.......... * ............ *
..... * ................. *
Q ------------------ R
PQ = √(1+x^2)
QR = x
PR = 1
∠PQR = θ
By this diagram, we can find tanθ = 1/x.
Next, we know
tan(π/2 - θ)
= sin(π/2 - θ)/cos(π/2 - θ)
= cosθ/sinθ
= 1/tanθ
So
tanθ = 1/tan(π/2 - θ)
We already know tanθ = 1/x, so
1/tan(π/2 - θ) = 1/x
x = tan(π/2 - θ)
arctan(x) = π/2 - θ
θ = π/2 - arctan(x)
And θ = arcsin(1/√(1+x^2)), so
arcsin(1/√(1+x^2)) = π/2 - arctan(x)
Therefore, C is a constant π/2.
