I got tricked by this

Geometry Level 3

Compute

$\sin(\cos^{-1}(\sin(\sin^{-1}(\cos(\sin^{-1}(-0.3)))))) .$

0.3

\sqrt{1-0.3^2}

-0.3

-\sqrt{1-0.3^2}

2 solutions

Brian Charlesworth
Nov 6, 2015

Note first that $\cos(\sin^{-1}(-0.3)) = \cos(-\sin^{-1}(0.3)) = \cos(\sin^{-1}(0.3)).$

Next, working from the middle of the expression outward, we then note that (i): $\sin(\sin^{-1}(y)) = y$ for $-1 \le y \le 1.$ As $y = \cos(\sin^{-1}(0.3))$ does lie in this interval, the given expression simplifies to

$\sin(\cos^{-1}(\cos(\sin^{-1}(0.3))).$

Now $\cos^{-1}(\cos(y)) = y$ for $0 \le y \le \pi.$ Since $y = \sin^{-1}(0.3)$ does lie in this interval, the expression further simplifies to

$\sin(\sin^{-1}(0.3)) = \boxed{0.3}$

due to statement (i) outlined above.

@Trevor Arashiro Did you go with $-0.3$ ? I always have to be very deliberate with expressions like these, hence my exposition being somewhat excessive. :)

Brian Charlesworth - 5 years, 7 months ago

Hahah, yes, I did. I simply glanced at every thing and since this was an easy contest, time was the only factor, so I never took a close look.

Trevor Arashiro - 5 years, 7 months ago

Aravind M
Dec 11, 2015

OMG! I did not notice there was a minus b4 0.3.... ;) Well you'll get the same answer if you proceed with 0.3. :D

I got tricked by this

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