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**Unformatted text preview: **nd sine of the quadrantal angles, but for
non-quadrantal angles, the task was much more involved. In these latter cases, we made good
use of the fact that the point P (x, y ) = (cos(Î¸), sin(Î¸)) lies on the Unit Circle, x2 + y 2 = 1. If
we substitute x = cos(Î¸) and y = sin(Î¸) into x2 + y 2 = 1, we get (cos(Î¸))2 + (sin(Î¸))2 = 1. An
unfortunate4 convention, which the authors are compelled to perpetuate, is to write (cos(Î¸))2 as
cos2 (Î¸) and (sin(Î¸))2 as sin2 (Î¸). Rewriting the identity using this convention results in the following
theorem, which is without a doubt one of the most important results in Trigonometry.
Theorem 10.1. The Pythagorean Identity: For any angle Î¸, cos2 (Î¸) + sin2 (Î¸) = 1.
The moniker â€˜Pythagoreanâ€™ brings to mind the Pythagorean Theorem, from which both the Distance
Formula and the equation for a circle are ultimately derived.5 The word â€˜Identityâ€™ reminds us that,
regardless of the angle Î¸, the equation in Theorem 10.1 is always true. If one of cos(Î¸) or sin(Î¸)
is known, Theorem 10.1 can be used to determine the other, up to a sign, (Â...

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