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The Pythagorean Theorem and the Unit Circle
This following diagram shows that $\displaystyle \sin^2 \theta + \cos^2 \theta = 1$, given $ \displaystyle \cos \theta \equiv \frac{adj}{hyp}$ and $ \displaystyle \sin \theta \equiv \frac{opp}{hyp}$ and the Pythagorean theorem. Here, $opp$, $hyp$, and $adj$ are variables for lengths. The values corresponding to these lengths are positive numbers or zero.
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Double Angle Formulas
In the Angle Addition Identities post it was shown that $ \displaystyle \sin(x+y) = \sin x \cos y + \sin y \cos x $ and $ \displaystyle \cos(x+y) = \cos x \cos y – \sin x \sin y $. These identities are valid if $\sin \theta \equiv \frac{opp}{hyp}$ and $\cos \theta \equiv \frac{adj}{hyp}$, which implies that $\theta…