Proofs, Identities, and Toolkit Functions
Appendix
Important Proofs and Derivations
Product Rule
Proof:
Letand
Write in exponent form.
and
Multiply.
Change of Base Rule
whereand
are positive, and
Proof:
Let
Write in exponent form.
Take theof both sides.
When
Heron’s Formula
where
Proof:
Let[latex]b,[/latex]and
be the sides of a triangle, and
be the height.

So.
We can further name the parts of the base in each triangle established by the height such that

Using the Pythagorean Theorem,and
Sincethen
Expanding, we find that
We can then addto each side of the equation to get
Substitute this result into the equationyields
Then replacingwith
gives
Solve forto get
Sincewe get an expression in terms of
[latex]b,[/latex]and
Therefore,
And sincethen
Properties of the Dot Product
Proof:
Proof:
Proof:
Standard Form of the Ellipse centered at the Origin
Derivation
An ellipse consists of all the points for which the sum of distances from two foci is constant:

Consider a vertex.

Then,
Consider a covertex.

Then
Let

Becausethen

Standard Form of the Hyperbola
Derivation
A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances between two fixed points is constant.

Diagram 1: The difference of the distances from Point P to the foci is constant:
Diagram 2: When the point is a vertex, the difference is
Defineas a positive number such that

Trigonometric Identities
Pythagorean Identity | ![]() |
Even-Odd Identities | ![]() |
Cofunction Identities | ![]() |
Fundamental Identities | ![]() |
Sum and Difference Identities | ![]() |
Double-Angle Formulas | ![]() |
Half-Angle Formulas | ![]() |
Reduction Formulas | ![]() |
Product-to-Sum Formulas | ![]() |
Sum-to-Product Formulas | ![]() |
Law of Sines | ![]() |
Law of Cosines | ![]() |
ToolKit Functions



Trigonometric Functions
Unit Circle

Angle | ![]() |
![]() |
![]() |
![]() |
![]() |
---|---|---|---|---|---|
Cosine | 1 | ![]() |
![]() |
![]() |
0 |
Sine | 0 | ![]() |
![]() |
![]() |
1 |
Tangent | 0 | ![]() |
1 | ![]() |
Undefined |
Secant | 1 | ![]() |
![]() |
2 | Undefined |
Cosecant | Undefined | 2 | ![]() |
![]() |
1 |
Cotangent | Undefined | ![]() |
1 | ![]() |
0 |