- Pythagorean Theorem
- Definitions of sin, cos, & tan based on a right triangle.
- The Unit Circle
- Law of Sines, Law of Cosines, and various trigonometric identities.
- Derivatives
- Taylor series

*Define*the trig functions in terms of Taylor series, prove some theorems using them, and

*eventually*connect it to geometry?

Define:

- S(x) = x - x
^{3}/3! + x^{5}/5! - x^{7}/7! + x^{9}/9! - ... - C(x) = 1 - x
^{2}/2! + x^{4}/4! - x^{6}/6! + x^{8}/8! - ... - E(x) = 1 + x + x
^{2}/2! + x^{3}/3! + x^{4}/4! + ...

- S(0) = 0
- C(0) = 1
- E(0) = 1
- S(-x) = -S(x)
- C(-x) = C(x)

S'(x) = d/dx (x - x

^{3}/3! + x

^{5}/5! - x

^{7}/7! + x

^{9}/9! - ...)

= 1 - 3x

^{2}/3! + 5x

^{4}/5! - 7x

^{6}/7! + 9x

^{8}/9! - ...

= 1 - x

^{2}/2! + x

^{4}/4! - x

^{6}/6! + x

^{8}/8! -

= C(x)

C'(x) = d/dx (1 - x

^{2}/2! + x

^{4}/4! - x

^{6}/6! + x

^{8}/8! - ...)

= 0 - 2x/2! + 4x

^{3}/4! - 6x

^{5}/6! + 8x

^{7}/8! - ...

= -x + x

^{3}/3! - x

^{5}/5! + x

^{7}/7! - ...

= -(x - x

^{3}/3! + x

^{5}/5! - x

^{7}/7! + ...)

= -S(x)

E'(x) = d/dx (1 + x + x

^{2}/2! + x

^{3}/3! + x

^{4}/4! + ...)

= 0 + 1 + 2x/2! + 3x

^{2}/3! + 4x

^{3}/4! + 5x

^{4}/5! + ...

= 1 + x + x

^{2}/2! + x

^{3}/3! + x

^{4}/4! + ...

= E(x)

Now, consider the function f(x) = S(x)

^{2}+ C(x)

^{2}. Using the Power Rule + Chain Rule, its derivative is:

f'(x) = d/dx (S(x)

^{2}+ C(x)

^{2})

= 2 S(x) S'(x) + 2 C(x) C'(x)

= 2 S(x) C(x) + 2 C(x) (-S(x))

= 2 S(x) C(x) - 2 S(x) C(x)

= 0

Thus, f is a constant function. To find the value of this constant, simply plug in an arbitrary x. Zero will do nicely.

f(0) = S(0)

^{2}+ C(0)

^{2}

= 0

^{2}+ 1

^{2}

= 0 + 1

= 1

So S(x)

^{2}+ C(x)

^{2}= 1 for any x. This might come in handy later, but for now I don't know what significance it may have.