- WELCOME TO A LESSON ON THE PERIOD OF A TANGENT
AND COTANGENT FUNCTIONS.
THE GOALS OF THE VIDEO ARE TO DETERMINE THE PERIOD
OF A FUNCTION AND ALSO TO GRAPH THE TANGENT
AND COTANGENT FUNCTIONS WITH VARIOUS PERIODS.
SO AS MENTIONED EARLIER, A FUNCTION, F,
IS SAID TO BE PERIODIC IF F OF X IS EQUAL TO F OF X + NP.
SO IF THESE TWO FUNCTIONS ARE EQUAL FOR ALL VALUES OF X,
THE LEAST POSSIBLE CONSTANT P IS CALLED THE PERIOD.
AND THIS PERIOD DOES HAVE TO BE POSITIVE.
A FUNCTION WITH PERIOD P WILL REPEAT
ON INTERVALS OF LENGTH P.
SO THE FUNCTION WE SEE HERE AFTER FIVE UNITS,
IT REPEATS ITSELF, AND THEREFORE THE PERIOD OF THIS FUNCTION
IS EQUAL TO 5.
FIRST OF ALL, THIS VIDEO DOES ASSUME
THAT YOU'RE COMFORTABLE WITH THE GRAPHS
OF THE BASIC TANGENT AND COTANGENT FUNCTIONS.
AND AS WE SEE HERE, FOR Y = TAN(X),
AFTER -PI OVER 2 TO PI OVER 2, THE FUNCTION REPEATS ITSELF.
AND THEREFORE THE PERIOD OF THIS FUNCTION IS PI RADIANS.
AND NOTICE IT'S THE SAME FOR A COTANGENT.
AFTER 0 TO PI RADIANS, THE FUNCTION
BEGINS TO REPEAT ITSELF.
SO THE GRAPH OF Y = TANGENT(BX) OR Y = COTANGENT(BX)
WILL HAVE THE PERIOD OF PI RADIANS DIVIDED BY B,
OR IF WE'RE TALKING ABOUT DEGREES,
180 DEGREES DIVIDED BY B.
SO IF WE WANT TO DETERMINE THE PERIOD OF Y = TAN(3X),
WE HAVE A VALUE OF B = 3.
SO THE PERIOD WILL BE PI DIVIDED BY B OR 3,
THEREFORE THE PERIOD IS PI OVER 3 RADIANS,
WHICH IS EQUAL TO 60 DEGREES OR 180 DEGREES DIVIDED BY 3.
SO IN RADIANS IT'S PI OVER 3. AND IN DEGREES IT'S 60 DEGREES.
FOR Y = COTANGENT (X DIVIDED BY 4),
YOU CAN THINK OF THIS AS 1X DIVIDED BY 4,
SO THE VALUE OF B IS 1/4.
SO IN RADIANS, IT WOULD BE PI DIVIDED BY 1/4,
WHICH IS PI TIMES THE RECIPROCAL OR PI x 4 OR 4PI.
AND THEN AGAIN, IN DEGREES, 4PI RADIANS
IS EQUAL TO 720 DEGREES.
SO, AGAIN, WE HAVE THE PERIOD IN BOTH RADIANS AND DEGREES.
THE AMPLITUDE OF A PERIODIC FUNCTION
MEASURES ITS HEIGHT.
THE AMPLITUDE IS DEFINED AS THE FARTHEST DISTANCE
THE WAVE REACHES FROM THE CENTER OF THE WAVE.
HOWEVER, SINCE TANGENT AND COTANGENT
DON'T HAVE MAXIMUMS OR MINIMUMS BECAUSE THEY CONTINUE UP FOREVER
AND DOWN FOREVER, THERE IS NO AMPLITUDE.
SO HERE IS THE FORM OF THE EQUATION
WE'RE GRAPHING RIGHT NOW.
B AFFECTS THE PERIOD, AND EVEN THOUGH TANGENT
DOESN'T HAVE AN AMPLITUDE, THE VALUE OF "A" WILL AFFECT
THE GRAPH SLIGHTLY, WHICH WE'LL DISCUSS SHORTLY.
LET'S START OFF BY FINDING THE PERIOD.
SO, REMEMBER, IT'S PI DIVIDED BY B.
IN THIS CASE, OUR B IS EQUAL TO 1/2.
PI DIVIDED BY 1/2 IS THE SAME AS PI
TIMES THE RECIPROCAL OR PI x 2, AND THAT WOULD GIVE US
2PI RADIANS FOR THE PERIOD.
THE BASIC TANGENT FUNCTION CAN BE GRAPHED
BETWEEN -PI OVER 2 AND PI OVER 2 RADIANS.
BUT SINCE THE PERIOD IS NOW 2PI, WE'LL EXTEND LEFT
AS FAR AS -PI RADIANS AND AS FAR RIGHT AS PI RADIANS.
AND WITHIN THIS INTERVAL, WE CAN GRAPH ONE COMPLETE PIECE
OF Y = 2TAN(X DIVIDED BY 2).
SO WE'LL HAVE VERTICAL ASYMPTOTES AT -PI
AND PI RADIANS NOW.
NOW, JUST AS GRAPHING THE OTHER TRIGONOMETRIC FUNCTIONS,
WE'RE GOING TO DIVIDE THIS INTERVAL
INTO FOUR EQUAL PARTS.
AND NOW GRAPH KEY VALUES OF THE FUNCTION.
SO, FIRST THING WE NOTICE IS THAT TANGENT OF 0 RADIANS
IS EQUAL TO 0.
AND THE SAME WILL BE TRUE FOR THIS FUNCTION.
SO WE CAN GRAPH, NOTICE ON THIS FOURTH OF THE GRAPH,
THE BASIC TANGENT FUNCTION IS EQUAL TO 1.
AND THE CORRESPONDING POINT ON THIS GRAPH WILL BE EQUAL TO 2
BECAUSE WHEREVER THE TANGENT FUNCTION
IS EQUAL TO 1, WE'RE GOING TO MULTIPLY IT BY 2.
AND ON THE FOURTH WHERE THIS FUNCTION
IS EQUAL TO -1, THIS FUNCTION WILL BE EQUAL TO -2.
AND THAT'S REALLY ENOUGH INFORMATION
TO MAKE A DECENT GRAPH OF THIS FUNCTION.
WE KNOW THEIR SHAPE WILL STAY PRETTY MUCH THE SAME.
SO IT'S GOING TO APPROACH THIS VERTICAL ASYMPTOTE
ON THE RIGHT AND THE SAME ON THE LEFT.
AND OF COURSE WE COULD CONTINUE THIS PATTERN
ON INTERVALS OF 2PI IF WE NEEDED TO.
SO, NOTICE THAT THIS VALUE OF "A"
DOESN'T AFFECT THE AMPLITUDE, BUT IT DOES KIND OF
VERTICALLY STRETCH THE BASIC TANGENT FUNCTION
ON THIS NEW PERIOD.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE.
WE HAVE AN "A" VALUE OF 1/2, WHICH, AGAIN,
IS NOT THE AMPLITUDE FOR THIS FUNCTION.
THIS FUNCTION DOES NOT HAVE AN AMPLITUDE.
OUR B VALUE IS EQUAL TO 2, SO OUR PERIOD
WILL BE PI DIVIDED BY 2 RADIANS.
WE NORMALLY GRAPH ONE COMPLETE PIECE
OF A COTANGENT FUNCTION FROM 0 TO PI RADIANS.
BUT NOW SINCE THE PERIOD IS PI RADIANS,
WE'LL HAVE A COMPLETE PIECE OF THE GRAPH
FROM 0 TO PI OVER 2 RADIANS.
SO WE'LL HAVE A VERTICAL ASYMPTOTE AT 0
AND PI OVER 2 RADIANS.
NEXT, WE'LL DIVIDE THIS INTO FOUR EQUAL PARTS.
HALF OF PI OVER 2 WOULD BE PI OVER 4.
HALF OF THAT WOULD BE PI OVER 8 AND SO THIS WOULD BE 3PI OVER 8.
NOW, LET'S GO AHEAD AND GRAPH SOME KEY POINTS
ON THIS FUNCTION.
LOOKING AT THE GRAPH OF Y = COTANGENT(X),
NOTICE ON THE INTERVAL EQUAL TO THE PERIOD,
AT THE HALFWAY MARK, THIS FUNCTION'S EQUAL TO 0.
AND IT WILL STAY THE SAME ON THIS GRAPH.
SO AT PI OVER 4, THIS FUNCTION'S EQUAL TO 0.
NOW, IN THE FIRST FOURTH, WHERE THIS FUNCTION
IS EQUAL TO 1, THIS FUNCTION WILL BE EQUAL
TO 1/2 SINCE IT'S 1/2 TIMES THE COTANGENT OF 2X.
AND ON THE THIRD FOURTH WHERE THIS FUNCTION
IS EQUAL TO -1, THIS FUNCTION WILL BE EQUAL TO -1/2.
AND THE GENERAL SHAPE WILL REMAIN THE SAME.
SO OUR FUNCTION WOULD LOOK SOMETHING LIKE THIS.
AND NOW WE COULD REPEAT THIS IF NEEDED ON INTERVALS
OF PI OVER 2.
WHAT I'D LIKE TO DO NOW IS GO BACK AND TAKE A LOOK
AT BOTH OF THESE ON THE GRAPHING CALCULATOR.
LET'S START BY TAKING A LOOK AT THIS ONE.
NOW, I'M IN RADIAN MODE.
LET'S GO AHEAD AND TAKE A LOOK AT THE WINDOW FIRST.
GIVEN THE PERIOD WAS 2PI RADIANS,
BUT WE KNEW WE'D HAVE ONE COMPLETE PIECE
OF THE FUNCTION FROM -PI TO PI, I ENTERED AN X-MIN OF -PI,
AN X-MAX OF PI, AND I SCALED THE Y-AXIS
FROM -5 TO 5.
NOW, WHAT I'M GOING TO DO IS I'M GOING TO MAKE THE GRAPH
OF THE FUNCTION WE'RE TRYING TO SKETCH BOLD.
SO I'M GOING TO GO TO THE LEFT AND PRESS "ENTER,"
AND THEN WE'LL GRAPH IT.
THERE'S A GRAPH OF THE BASIC TANGENT FUNCTION.
AND THERE'S A GRAPH OF Y = 2TAN(X DIVIDED BY 2).
AND NOTICE THAT IT'S CHANGED THE PERIOD
AND ALSO VERTICALLY STRETCHED IT SLIGHTLY.
LET'S GO AHEAD AND TAKE A LOOK AT OUR SECOND EXAMPLE.
SO WE'LL GRAPH THE PARENT FUNCTION FIRST.
Y = COTANGENT(X), REMEMBER, THAT'S GOING TO BE
1 DIVIDED BY TAN(X) 'CAUSE THEY'RE RECIPROCALS.
NEXT, I'M GOING TO GRAPH 1/2 x 1/TAN(2X),
SO I'LL HAVE 1/2 IN PARENTHESES TIMES 1 DIVIDED BY TAN(2X).
AND LET'S GRAPH THEM.
AGAIN, THE ORIGINAL FUNCTION IS THIN.
AND THE FUNCTION WE WANT TO GRAPH IS IN BOLD.
NOTICE THE PERIOD HAS OBVIOUSLY CHANGED
'CAUSE IT'S ONLY PI OVER 2 RADIANS NOW
AND IT HAS BEEN SOMEWHAT VERTICALLY COMPRESSED
EVEN THOUGH IT'S HARD TO TELL BECAUSE THE PERIOD HAS CHANGED.
OKAY, THAT VERIFIES OUR WORK.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.