Many approximations!

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Given that s e c x + t a n x ≈ 1 − l n ( 1 − x ) secx + tanx \approx 1 - ln(1-x) , and that s e c x + t a n x = 3 2 secx + tanx = \frac{3}{2} , the approximate value for the principle value of s i n − 1 5 13 sin^{-1}\frac{5}{13} can be expressed as s i n − 1 5 13 ≈ A + B e C sin^{-1}\frac{5}{13} \approx A + Be^C , find the value of − 10 ( A + B + C ) -10(A+B+C) .


The answer is 5.

Tan Kiat by Tan Kiat , 24, Singapore

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1 solution

Patrick Corn
Jan 9, 2014

I think this problem would be better if you omitted the part about s e c ( x ) + t a n ( x ) = 3 2 {\rm sec}(x) + {\rm tan}(x) = \frac32 ; given that x = sin ⁡ − 1 5 13 x = \sin^{-1} \frac5{13} , you can compute this directly.

At any rate, we set 3 2 = 1 − ln ⁡ ( 1 − x ) \frac32 = 1 - \ln(1-x) and solve to get x = 1 − e − 1 / 2 x = 1-e^{-1/2} , so the solution is − 10 ( 1 − 1 − 1 / 2 ) = 5 -10(1-1-1/2) = \fbox{5} .

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