Trig Functions and Mind Mapping

Note: Sin (x) always mirrors the x axis while Cos (x) always mirrors the y axis.

Example: Cosecant

Example: Secant

Hills and Valleys

When compared to the sine cosine functions, the max. (hill) and min. (valleys) are the opposite of the cosecant secant functions. In other words, the sine max.

corresponds to the cosecant's minimum valley, as shown on the right.

Example: Cotangent

Secant and Cosecant Functions

Cotangent Function

It wasn't until the early 60s or 50s that a theory affiliated with mind mapping was introduced (network semantics) explaining the process of human learning.

Tangent Function

Example: Tangent Graph

The 5 Guidelines for Verifying Trig Identities

1. Work with ONE SIDE of the equation at a time

2. Look for OPPORTUNITIES to FACTOR an expression, ADD fractions, SQUARE a binomial, or CREATE a monomial denominator

3. Look for OPPORTUNITIES to USE the fundamental identities

4. If the latter fail, try converting all terms to sines and cosines

5. Always try SOMETHING

Law of Cosines

Graphing Tan, Cot, Sec, and Csc

History of Mind Mapping

LAW of Sines

By 3rd BC, many well known philosophers

were thought to have used mind mapping

as a form of getting thoughts on paper using words and pictures. Evidently, Porphyry of Tyros was the first to use mind mapping, then followed a couple other philosophers, Ramon Llull and Leonardo da Vinci.

Sum and Difference Formula

The main purpose of mind mapping -

as many would agree - is to allow

people of all ages to brainstorm in a

way that allows them to trace their

ideas to broader or more specific

concepts. This way, mind mapping

allows one to visualize an entire

thought process, making it easier to elaborate on the specifics in a more manageable way.

Power-Reducing Formulas

Double-Angle Formulas

Half-Angle Formulas

Product to Sum Formulas

Additional Formulas

Sum to Product Formulas

The Right Triangle Perspective: Trig Functions and Identities

Ross Quilian and Alan Collins are known as

fathers of mind mapping because they were

able to expand on the structure of human

learning and connecting major ideas to ideas.

Trig Functions and

Mind Mapping

Using various combinations of the three sides of triangles (named above), it is possible to find the six trig functions of the acute angle theta.

The Trigonometric Functions

When you don't know a side or hypotenuse, you can use the Pythagorean theorem:

to solve for either side and plug it in

Confunction Identities

The Fundamental Trigonometric Identities Cont'd

Example: 3

Along with the three FTI that were mentioned before, where theta becomes "u", there are two more categories:

Example: 4

- Confunction Id.

- Even/Odd Id.

The Reciprocal Identities

Special Angles

The Fundamental Trigonometric Identities

Original Prezi by Louisa Montaño for Mr. Gentry's A block PreCal Class

Even/Odd Identities

The fundamental trig identities are divided into three parts:

The Quotient Identities

Example: 5

- Reciprocal Id.

- Quotient Id.

- Pythagorean Id.

The Pythagorean Identities

The Unit Circle

A Summation of the Inverse Functions

Other Inverse Functions:

Cosine

Inverse Tangent Function

Inverse Trig Functions: Composition of Functions

Graphing Sin and Cos

Here are the basic characteristics of the parent sin and cos functions:

Example: Inverse Sine Function

Sine function:

Functions with inverses must:

- pass the horizontal line test anytime, or..

- when it is restricted to points

Example: 7 Cosine

When you graph the sin and cos of x, you need to keep five key points in mind; the intercepts, maximum point, and minimum point.

Example: 6 Sine

when you divide period 2 pi by four, you get..

Furthermore...

The Trig Identities

Given the trig function identities, to find the trig functions that correspond to a point on the unit circle, we must plug the coordinate values (x, y), into the individual trig identities.

Example: 1

Example: 2