sin[tan^-1(-8)]

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sin[tan^-1(-8)]

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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Is tan^-1(-8) the arctan of -8 ?
Suppose you let \(x=\tan^{-1}(-8)\). This would mean that \(\tan x=-8\). Keep in mind that \(\tan x\) is defined for \(\left(\dfrac{(2n+1)\pi}{2},\dfrac{(2n+3)\pi}{2}\right)\) for integers \(n\). Its inverse is typically defined by restricting the domain of \(\tan x\) to \(\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\) (i.e. the first and fourth quadrants of the unit circle). In terms of the quadrants, \(\tan x\) is negative for values of \(x\) within the second and fourth quadrants. Draw up a reference triangle with angle \(x\) such that \(\tan x=-8=\dfrac{-8}{1}\): |dw:1439434906616:dw| If you can find the missing side, you can determine \(\sin x=\sin\left(\tan^{-1}(-8)\right)\).

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