### Precalculus

#### Welcome to Students

- What does this math course have to offer?
- Precalculus builds on your algebra foundation to explore further
implications of the
*function* concept, starting with graph
transformations of the functions we already know. We draw on geometry
knowledge to develop the *circular functions*, which enjoy a beautiful
connection to the exponential function by way of the complex numbers and
infinite series. Armed with these tools, we then consider polar coordinates,
*vector-valued functions* (AKA parametric equations), and the
algebra of vectors in Cartesian coordinates.
- What are some skills that I can take from this class?
**Strengthened algebraic manipulation** emerges as you work
through proofs of identities and simplification of transcendental
expressions. This fluency will remove for you one of the major obstacles to
success in calculus.
**Deepened mathematical appreciation** follows from
revisiting the same result in different settings, e.g. proving
the addition formulas for sine and cosine by using a coordinate
transformation, or by multiplying two exponential functions of imaginary
arguments and interpreting the result in both polar and rectangular
coordinates.
**Expanded mathematical vocabulary** will allow you to talk
about periodic phenomena with circular functions and to model Euclidean
geometry problems with vectors.

- What do I need to succeed in this class?
- Just as you would for any class you take in college, be sure to read the
assigned text before each lecture. Make sure you
*read actively*,
with pen or pencil in hand and a dedicated math notebook ready for jotting
down the key ideas. Adopt a skeptical attitude toward the expositions
provided, coming up with questions that force the words on the page to
reveal meanings and connections buried under the surface. Play with math as
if you were a toddler trying a new toy, looking for all the ways it can
be broken or repurposed. After you think you understand a new concept from
all possible angles, attempt to solve the end-of-section exercises
without referring to the example problems in the text.

#### Further Thoughts on My Precalculus Teaching Philosophy