**Trigonometric Functions**

**Modeling Representation**

Trigonometric functions are sometimes called **circular functions**. This is because the two fundamental trigonometric functions – the **sine** and the **cosine** – are defined as the coordinates of a point P travelling around on the **unit circle** of radius 1 .

As P moves around the circle of radius 1, the input variable – the angle q – changes. The outputs –

Methods for **finding sines & cosines**

This gives us a clue about the kinds of phenomena that trigonometric functions are likely to model. Going around in a circle is a very simple kind of **periodic** behavior. Periodic behaviors repeat themselves at regular intervals. The sine and the cosine repeat their *outputs* at regular intervals. Indeed, each time P travels once around the circle (the input q changes by 2p radians or 360º) the coordinates of P (the outputs of sine and cosine) repeat.

By changing the size of the circle and the speed at which the point P travels around it, transformations of the sine and the cosine can be made to model a wide variety of regularly repeating behaviors.

Trigonometric functions model data with |

Here are some situtations that should make you think of trigonometric functions:

- A ferris wheel. (horizontal and vertical distance vs. angle)

- A hula dancer. (horizontal and vertical distance vs. angle)

- The length of daylight. (length vs. day of the year)

- The changing seasons. (temperature vs. day of the year)

- A repetitious calculation. (step vs. time)

Words and phrases like "seasonal," "repeatedly," and "over and over" usually indicate the presence of trigonometric functions.