Can anyone help with some trigonometry (periodic functions)?

Question

I don't understand how to use periodic functions. I've read the chapter over and over but it just doesn't make since to me. I was wondering if in maybe someone elses words it would make more since.

Here is an example question from the problems that confuse me:

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator.

1.) tan 405 degrees

and also

2.) cos 33pi/4

Could someone please explain how to go about these problems in simplest terms possible? Please and Thank You.

Answer 1

You must know the four-quadrant model for the angles from 0 to 360 degrees, where the radius arm starts at the 3-o-clock position (0 degrees) and rotates in an anticlockwise direction?

A full rotation of the radius arm is 2π radians, or 360 degrees, so for any angle greater that that, the trigonometric functions just repeat.

With 405 degrees, therefore, you just subtract 360 to get 45 degrees; so tan 405 degrees is the same as tan 45 degrees.  You could add another 360 degrees to get 765 degrees, and the sin, cos, tan, functions are still the same (as the values for 45 degrees.)

If you are measuring angles in radians, the full rotation is 2π, so the functions repeat after 2π,(or 8/4 if you are measuring in π / 4 steps).

Or, in other words, (33 / 4)π  =  8¼ π, so you can subtract 2π four times to get just ¼ π; so tan (33 / 4)π  is the same as tan π /4 Which is equivalent to 45 degrees.