Level 8: Proof, radians and vectors

Geometry · level 8 of 8 · rough age 17 and beyond (a guide for parents; anyone can start anywhere)

The last level of geometry is where it meets university mathematics. It starts with proof, the way mathematicians become certain, and then frees sine and cosine from triangles: angles measured in radians, the unit circle, and the waves it draws. Vectors turn pictures into calculations, the dot product finds angles in any number of dimensions, and the final lesson steps off the flat page onto a sphere, where triangles can have three right angles.

Lesson	You will be able to
1. How to write a geometric proof	Write two-column and paragraph proofs, use congruence, and prove the triangle angle sum and the isosceles triangle theorem
2. Radians and the unit circle	Measure angles in radians, find exact values of sine and cosine for any angle, and prove identities such as \(\sin^2\theta + \cos^2\theta = 1\)
3. Graphs of sine and cosine	Sketch and read \(y = A\sin(B(x - C)) + D\), and model wheels, tides and daylight with it
4. Vectors: adding, scaling, length, components	Add, subtract and scale vectors, switch between components and length-and-direction, and prove facts with vectors
5. The dot product and projections	Find the angle between vectors, test for perpendicular, project one vector onto another and compute work
6. Beyond Euclid: geometry on a sphere	Use great circles and Girard's theorem, compute flight distances on Earth and Mars, and describe non-Euclidean geometry

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