Taken from my answer to “Proving a weird trig identity” at Math.StackExchange.com . $$\Large{\frac{\cos\theta}{1-\sin\theta} = \frac{\sec\theta + \tan\theta}{1}}$$ In the diagram, \(\overline{AB}\) is tangent to the unit circle at P. The “trig lengths” (except for \(|\overline{AQ}|\)) should be clear. We note that \(\angle BPR \cong \angle RPP^\prime\), since these inscribed angles subtend congruent arcs \(\stackrel{\frown}{PR}\) […]
Archive for the ‘Classroom’ Category
Identity from Math.StackExchange.com
Product Identities
$$\begin{align} 2 \cos A \cos B &= \cos(A-B) \;+\; \cos(A+B) \\ 2 \sin A \;\sin B &= \cos(A-B) \;-\; \cos(A+B) \end{align}$$
Proof Without Words: Nine-Point Circle Property
The “Nine-Point Circle” of a triangle \(\triangle ABC\) contains the vertices of the medial triangle (\(\triangle RST \)) and orthic triangle (\(\triangle UVW \)), as well as points \(X\), \(Y\), \(Z\) that bisect segments from \(\triangle ABC\)’s orthocenter (\(H\)) to its vertices. A question on Math.StackExchange.com asked for proof that the center of the nine-point circle […]
Limits are about the journey …
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A Ceva-like Theorem for Tetrahedra
A question at Math.StackExchange asked about “Ceva’s Theorem in three dimensions”. So I derived one. The result (which I believe may be new, although it could very well exist in the hundreds of years of mathematical literature since Ceva) replaces the traditional ratios of segment lengths with “triple-ratios” of triangle areas, to satisfying effect. Whereas […]