This book provides clear, detailed descriptions of how formal, abstract mathematics works. It takes you step-by-step through how to read definitions and construct formal proofs. It will also give you a brief introduction to analysis, abstract algebra, and logic. This book should be read in linear order: random access will not work well.
Can you reproduce the quadratic formula from memory? How about the formula for the tangent of half an angle? No? This isn't surprizing: it's probably been years since you last used them. You should own a book of mathematical formulas.
There are many books of mathematical formulas, in all sizes and prices, some intended for mathematicians and some for students in the physical sciences. Examine the collections at your local library and bookstores and buy one that you like. It should include trig formulas, formulas for volumes and areas of regular solids and parts of them (e.g. subsections of a sphere), integral and derivative tables, and a table of values for the integral of the Gaussian (or normal) distribution.
The CRC Press (CRC originally stood for "Chemical Rubber Company") has been publishing such handbooks, including not only several sizes of math handbooks, but also a giant handbook for chemists and physicists. Their big math handbook is
This costs about $40 new (hardbound). Used copies often appear in used bookstores. Make sure a used copy is recent enough that it does not include many pages of trig and log tables, redundant now that calculators are cheap.
They also publish a tiny math handbook: cheaper and fits in your backpack. It is adequate for most purposes; your library should have a copy of the larger handbook if you need to look up something obscure.
Also, a variety of useful formulas can be found in chapters 2 and 3 of