# 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

**Heronβs Formula**

where

**Proof:**

Let[latex]b,[/latex]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[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 |