7.7.misc-eg-1 - Example Simplify: cos tan−1 x , x ∈...

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Unformatted text preview: Example Simplify: cos tan−1 x , x ∈ (−∞, ∞). Because x is arbitrary in (−∞, ∞), we will not be able to find θ ∈ (−π/2, π/2) for which tan θ = x. A better approach is to look at the right triangle inscribed in a circle of radius r associated with such an angle θ. The information tan θ = x = x 1 tells us that such a triangle has vertical leg x, horizontal leg 1, and hypotenuse (also the radius √ r of the circle) given by r = 1 + x2 . We were not asked for the value of θ = tan−1 x, but for the value of cos θ. Looking at the two right triangles, we see that we get the same answer in either case: cos(tan−1 x) = cos θ = adjacent side 1 =√ , hypotenuse 1 + x2 x ∈ (−∞, ∞). ...
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