Reddit reviews Combined Answer Book For Calculus Third and Fourth Editions

FYI, there's an answer book available for Spivak's Calculus that is very useful for self-study students.

Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).

Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.

Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.

You could try "Precalculus" by Stitz & Zeager. Chapters 10 and onwards is their Trigonometry book. This should be a very smooth book to work through.

Have you already picked out a Calculus textbook? Also, what are her plans as an MIT student? If she's going into engineering and the like, I would say Larson's "Calculus" (solutions manuals vol 1, 2) would be good enough.

If she plans on being a math student, though, I would say give her a a few months with Velleman's "How To Prove It". Afterwards, I can't recommend Spivak's Calculus (Answer Book) and Jim Hefferon's Linear Algebra (solutions manual on same page) enough. This is a good time to introduce mathematical rigor as a normal thing in mathematics because, really, this is what math is about.