Proof of the Triangle Inequality for Real Numbers

The triangle inequality for real numbers is $ |a+b| \le |a| + |b|$ in which $a$ is a variable for a real number, and $b$ is a variable for a real number. Proof:…

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Do the angle addition identities only work for positive angles?

The Angle Addition Identities were derived using $\sin\theta \equiv \frac{opp}{hyp}$ and $\cos\theta \equiv \frac{adj}{hyp}$. In addition, $\theta$ was constrained to the interval of $[0, \frac{\pi}{2})$. The question addressed in this post is, do the Angle…

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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, $opp$,…

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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…

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