Proofs, Identities, and Toolkit Functions

Appendix

Important Proofs and Derivations

Product Rule

Proof:

Letand

Write in exponent form.

and

Multiply.

Change of Base Rule

whereandare positive, and

Proof:

Let

Write in exponent form.

Take theof both sides.

When

Heronβs Formula

where

Proof:

Let$b,$andbe the sides of a triangle, andbe 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

SincethenExpanding, we find that

We can then addto each side of the equation to get

Substitute this result into the equationyields

Then replacingwithgives

Solve forto get

Sincewe get an expression in terms of$b,$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

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