Periodic Functions help (trigonometry)?

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

The values of all the trigonometric ratios for angle x and angle x + 360 deg. or x + 2pi rad. are always equal.

For tangent and cotangent, the ratios are equal also for angles x and x + 180 deg, or x + pi rad.

If your book doesn't show the graphs, look at them here:

http://www.themathpage.com/atrig/graphs-...

That site deals with sin, cos and tan. For cot, sec and cosec, you can plot the graphs yourself here:

http://www.walterzorn.com/grapher/graphe...

1.

tan(405)

Because the values of tan repeat every 180 deg, you can reduce the angle by 360 to 45 deg. without changing the value of the tangent.

As a 45-45-90 triangle is isosceles, tan(45) = 1, and therefore tan(405) = 1.

2.

cos(33pi / 4)

Removing multiples of 2pi, 33pi / 4 reduces to pi / 4.

cos(33pi / 4) = cos(pi / 4).

From the same isosceles triangle, if the legs are length 1, the hypotenuse is sqrt(2), and cos(pi / 4) = 1 / sqrt(2) = sqrt(2) / 2.