now let's talk about the periodic properties of trigonometric functions sine of theta is equal to sine of theta plus 360 degrees or 2 pi so what this means is that if you add 360 to theta the sine will have the same value so whenever you have the sine of an angle and a sign of another coterminal angle they will share the same value if you recall to find a coterminal angle you have to add or subtract by 360 so sine of 30 and sine of 390 have the same value because they differ by 360 now this equation is true for all of the trigonometric functions cosine tangent cotangent it's all true for each and every one of them so cosine of 50 is going to be equal to cosine of 50 plus 360 or cosine of 410 cosine of pi is equal to cosine pi plus 2pi or cosine 3pi so the word periodic means things that repeat if I add another 2 pi and get 5 pi cosine 5 pi will equal 2 all of these values when you graph the sine and cosine functions this is the graph of sine it repeats itself over and over so the value will be the same if you keep adding 360 to the angle now let's apply what we've learned into a problem let's say if we wish to evaluate sine of 420 degrees using periodic properties so we got to find a coterminal angle that's less than 420 so let's subtract 420 by 360 420 minus 360 is 60 so sine of 420 is equal to sine of 60 and based on a unit circle at 60 degrees we have the point and one-half comma root 3 over 2 and sine is equal to the Y value in a unit circle so sine 60 is root 3 over 2 which means that sine 420 is also equal to the square root of 3/2 now let's work on some more examples go ahead and find the value of cosine negative 2 PI over 3 so cosine negative 2 PI over 3 is equal to cosine negative 2 PI over 3 plus 2 PI and 2 pi is the same as 6 PI over 3 so therefore cosine negative 2 PI over 3 is equal to cosine 4 PI over 3 so his 4 PI over 3 and here's PI over 3 PI over 3 is the reference angle of 4 PI over 3 it's also the reference angle of negative 2 PI over 3 so at PI over 3 or 60 we have the point 1/2 root 3 over 2 now at 4 PI over 3 the X and the y values are negative cosine is equal to the x value so cosine 4 PI over 3 is negative 1/2 which is equal to cosine of negative 2 PI over 3 let's try one more example sine of 750 degrees go ahead and evaluate that one so using periodic properties let's bring the angle down let's subtracted by 360 750 - 360 is 390 so sign of 750 is equivalent to sign of 390 now 390 still high so let's subtract that by 360 one more time and so we're gonna get sine 30 so sine 30 sine 390 sine 750 they all equal each other and using the unit circle we know that sine 30 is 1/2 so therefore sine 750 is also 1/2