## Trigonometry and Trig Functions

#### Definitions and Graphs

The definitions of the Trigonometric Ratios are presented, along with some related exercises at the section on Trigonometric Functions at the University of Saskatchewan's "Exercises in Math Readiness" (EMR) site. The relationship between trig functions and circular motion is further explored in the preparatory section of the UBC on-line calculus course. For this, as well as many other properties of the trig functions, you might also check out the "ManipulaMath" trigJAVA applets created by the IES group in Japan. At y = sin x, a JAVA applet traces out the sine curve as a point moves around the unit circle, and similar applets deal with y = cos x. as well as other examples such as y = a sin b(x - c), and y = sin x + cos x. And a demonstration of the unit circle definitions of all six trig functions is another one of their many wonderful applets.

You can also use our own Graph Explorer to produce the graphs of various trig functions. (Any Computer Algebra System or Graphing Calculator would do as well of course but the detailed instructions might be a bit different.)

The effect of multiplying a basic trig function by another function to give a variable amplitude or "damping" effect is discussed at this page on the "CoolMath" site which also includes a page of other links to sites dealing with trigonometry and trig functions.

Although the trig functions are not one-to-one and so do not have inverse functions, they do have invertible restrictions and these are discussed at the UBC site on Inverse Trigonometric Functions

#### Identities and Equations

See Identities&Equations, at EMR.
Simple trigonometric equations are also dealt with at Trigonometric Equations, from the University of Texas El Paso's S.O.S. MATHematics site.

#### Applications

The EMR site includes this section on Applications of Trigonometric Functions.
Eric Hiob's collection of math applications at BCIT includes several involving trigonometry:
How high is that tree?,
another application to surveying ,
an application to Prosthetics and Orthotics,
and one to Fluid Flow in a Pipe .
Trigonometry and Music(@ArkansasStU)
Listening to equations(@ArkansasStU)
TidesExample@rice
Mercator@upenn

Buffon's Needle

#### Trigonometry and Complex Numbers

For the relationship between trig functions and complex exponentials
z  =   a + bi  =  r(cos q + i sin q)  =   reiq(Euler's formula),

See The Complex Plane,   Polar Form of a Complex Number : The unit circle,
and Euler's Formula from S.O.S. MATHematics.

#### Calculus with Trig Functions

The ManipulaMath trigJava applets also include a demonstration of the limit as x->0 of sin x/x .
Here's another version (of the same thing) as well as some other interesting graphics.

A different animation of the sine function, together with its gradient and derivative, can be seen at Calculus Animations.

#### Supplements, OnLine Courses, and More

There is a booklet of supplementary notes and practice problems on sale at the bookstore.

The University of Saskatchewan's "Exercises in Math Readiness" site has these notes and exercises on Trig Functions, Circular Functions, Identities&Equations, and Applications.
Simple trigonometric equations are also dealt with at Trigonometric Equations, from the University of Texas El Paso's S.O.S. MATHematics site.

For other online material you might try this short course in trigonometry , or (especially in preparation for Math174) The Trigonometric Functions and Calculus for Liberal Arts and Business Majors, a text resource written by Stefan Waner and Steven Costenoble of Hofstra University, and OregonState's "CalculusQuest" includes a precalculus "Field Guide" to functions which includes this section on trig functions, which has been reviewed by Frank Choy.

You might also check our 'raw list' (of links provided without comment) to see if there are any
more examples there that we haven't yet included here.

If you have come across other good web-based illustrations of trigonometry concepts,