Trig and Lengths
The trig-bash unit. Several traditional-style geometry problems that are meant to be solved by chasing lengths and ratios, with the aid of trigonometric techniques.
Unit Summary
Have you ever felt that standard geometry techniques just weren't enough? You chase angle after angle, but at some point you realize you're not getting anywhere with angles alone. You need something more concrete...
Welcome to Trigonometry and Lengths - the unit that takes high school algebra and turns it into your best friend... but also your WORST NIGHTMARE!!!
Based off of a handout by Ankit Bisain, this unit uses the tools of trigonometry to compute exact lengths and solve for impossible angles. The two stars of the show will be the (Extended) Law of Sines and the Law of Cosines. Other key players include the Ratio Lemma, Trig Ceva, and some other trigonometric identities. Length formulas such as Stewart's Theorem and Ptolemy's Theorem/Inequality will also be prevalent in this unit.
This unit is a good precursor to the Linear Power unit, which is a unit that uses length computations and applies them in Power of a Point contexts.
Pros
Most geometry problems will ask you to (or involve having to) find a certain angle or prove a certain quadrilateral is cyclic. However, sometimes angle chasing is not enough, because the conditions that fix the missing angle you need are length conditions. This is why trigonometry is useful - it relates lengths and angles, and can be used when the former is more tractable than the latter.
Cons
However, because there is no way to check your work (unlike, e.g., homogeneity for Complex and Barycentric bashing), Trig bashing is much more prone to error. Additionally, some expressions that look unsimplifiable may actually reduce to a hidden, simpler form. For example, it turns out that $\cos^2 A + \cos^2 B + \cos^2 C + 2\cos A \cos B \cos C$ is simplifiable; can you guess what it simplifies to? (The answer is inside this unit.)
Notable Problems
2009 IMO 4: An interesting problem about angle bisectors and having to extract the possible values of an angle.
2024 USATST 2: A tricky problem where you're given angle conditions and have to prove an angle is exactly $45^{\circ}$.