A New Function

Geometry Level 3

Let A r c T r i g ( x ) = sin ⁡ − 1 1 x + cos ⁡ − 1 1 x + tan ⁡ − 1 x + csc ⁡ − 1 x + sec ⁡ − 1 x + cot ⁡ − 1 x . ArcTrig(x)=\sin^{-1}\frac{1}{x}+\cos^{-1}\frac{1}{x}+\tan^{-1}x+\csc^{-1}x+\sec^{-1}x+\cot^{-1}x. What is A r c T r i g ( 4 5 + 1 ) ArcTrig\left(\frac{4}{\sqrt{5}+1}\right) in degrees?


The answer is 270.

Trevor B. by Trevor B. , 22, USA

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1 solution

Anish Puthuraya
Feb 16, 2014

Since all the angles are within the domains of the respective inverse trig functions, thus,

sin ⁡ − 1 1 x + cos ⁡ − 1 1 x = π 2 \displaystyle \sin^{-1}\frac{1}{x}+\cos^{-1}\frac{1}{x} = \frac{\pi}{2}

tan ⁡ − 1 x + cot ⁡ − 1 x = π 2 \displaystyle \tan^{-1}x+\cot^{-1}x = \frac{\pi}{2}

csc ⁡ − 1 x + sec ⁡ − 1 x = π 2 \displaystyle \csc^{-1}x+\sec^{-1}x = \frac{\pi}{2}

Thus,
A r c T r i g ( 4 5 + 1 ) = π 2 + π 2 + π 2 = 3 π 2 = 27 0 o \displaystyle ArcTrig(\frac{4}{\sqrt{5}+1}) = \frac{\pi}{2}+\frac{\pi}{2}+\frac{\pi}{2} = \frac{3\pi}{2} = \boxed{270^o}

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