Periodic Functions (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

A periodic function means it repeats every period. cos and sin repeat every 360 degrees (2pi radians) and tan repeats every 180 degrees (pi radians).

1) since tan repeats every 180 degrees you can freely sunbtract or add 180 degrees to the number being 'tanned' and it will give the same answer:

tan 405 = tan 225 = tan 45 (which you probably know) = 1

2)cos repeats every 2pi radians (as this question is in radians) so similiarly you can freely add or subtract 2pi from the value being 'cossed'.

cos33pi/4 = coc 8.25pi = cos 0.25pi (which you probably know) = 1/√2

Answer 2

1) periodic means the function has the same value over and over again at a certain periodicity...trig functions repeat their values every 360 degrees, this means that if you draw the curve sin x (or cos x or tanx) you would get the same form of the curve over and over again once you got past 360 degrees on the x axis...

so, if you go around the circle 405 degrees, that means you have gone around one full revolution of 360 plus an additional 45 degrees

this means that tan(405)=tan45 which is 1

2) 360 degrees is 2 pi radians, so in this case you are told that you have gone around 32pi/4 +1pi/4 = 8pi +pi/4 radians

you know that 8 pi radians is just four times around the circle, so 33pi/4 means going around 8 times plus pi/4 additional

if there are 2pi radians in a circle, then pi/4 is 1/8 of  a circle, or 45 degrees, so

cos(33pi/4)=cos(pi/4)=cos(45)=sqrt(2)/...

Answer 3

Tan and cot "repeat" every 180 degrees and sin and cos every 360 degrees. That is what periodic means. Sin 30 = sin 390 for example.

tan 405 = tan 45 by subtracting 180 twice

tan 45 = 1 exactly.

cos 33pi/4 = 33(180)/4 = 1,485 degrees = 45 degrees subtracting 360 repeatedly. cos 45 = sq rt 2 /2 exactly.