- WELCOME TO A LESSON
ON THE AMPLITUDE AND PERIOD OF THE SINE AND COSINE FUNCTIONS.
THIS IS A SERIES OF THREE VIDEOS.
THE NEXT VIDEO WILL TAKE A LOOK
AT HORIZONTAL AND VERTICAL TRANSLATIONS,
AND THE THIRD WILL COMBINE ALL FOUR TRANSFORMATIONS IN ONE.
SO THE GOALS OF THIS VIDEO ARE TO DETERMINE
THE PERIOD OF THE FUNCTION,
AND THEN ALSO TO DETERMINE
THE AMPLITUDE OF THE SINE AND COSINE WAVE.
A FUNCTION F IS SAID TO BE PERIODIC
IF F OF X = F OF X + NP FOR ALL VALUES OF X.
SO IF THESE TWO FUNCTIONS ARE EQUAL
THEN THE LEAST POSSIBLE CONSTANT P IS CALLED THE PERIOD,
AND THE PERIOD IS REQUIRED TO BE POSITIVE.
A FUNCTION WITH PERIOD P WILL REPEAT ON INTERVALS OF LENGTH P.
SO IF YOU TAKE A LOOK AT THIS FUNCTION HERE,
IF WE START HERE WHEN X = 0, AND THEN WE GO OVER TO X = 5,
THIS IS ONE COMPLETE CYCLE OF THE FUNCTION
BECAUSE AFTER FIVE IT STARTS TO REPEAT ITSELF.
AGAIN, THE PERIOD IS THE HORIZONTAL DISTANCE NEEDED
BEFORE THE VALUES BEGIN TO REPEAT.
SO IF WE LOOK AT THE BASIC SINE FUNCTION WE CAN SEE
THAT AFTER THE HORIZONTAL DISTANCE 2PI
THE FUNCTION STARTS TO REPEAT ITSELF,
THEREFORE THE PERIOD = 2PI FOR SINE THETA.
AND IF WE TAKE A LOOK AT COSINE THETA
THE SAME THING OCCURS.
AFTER A HORIZONTAL DISTANCE OF 2PI
THE FUNCTION WOULD START TO REPEAT ITSELF AGAIN.
SO THE PERIOD OF THE BASIC SINE AND COSINE FUNCTION = 2PI.
HOWEVER, IF WE TAKE A LOOK
AT THE GRAPHS OF Y = SINE OF BX OR Y = COSINE BX,
THE PERIOD WILL CHANGE BASED UPON THE VALUE OF B.
AND, IN FACT, THE PERIOD WILL BE EITHER 360 DEGREES DIVIDED BY B
OR 2PI DIVIDED BY B
BASED UPON WHETHER WE'RE GRAPHING IN DEGREES OR RADIANS.
CHANGING B RESULTS IN WHAT IS CALLED A HORIZONTAL DILATION,
WHICH CAN ALSO BE CALLED
A HORIZONTAL STRETCH OR COMPRESSION.
SO IF WE WANTED TO DETERMINE THE PERIOD
OF Y = COSINE 3X IN RADIANS,
WE WOULD TAKE 2PI AND DIVIDE BY THE VALUE OF B, WHICH IS 3.
THIS IS OUR PERIOD IN RADIANS. IF WE WANT IT IN DEGREES,
WE COULD TAKE 360 DEGREES DIVIDE BY 3, WHICH IS 120 DEGREES.
AND OF COURSE 3PI RADIANS = 120 DEGREES.
FOR Y = SINE X DIVIDED BY 4,
YOU MIGHT WANT TO THINK OF THIS AS 1X/4.
SO THE VALUE OF B IS ACTUALLY 1/4.
2PI DIVIDED BY 1/4,
REMEMBER IS THE SAME AS 2PI x THE RECIPROCAL,
SO x 4/1 OR JUST 4, WHICH IS EQUAL TO 8PI.
AND THIS IS EQUIVALENT TO 1,440 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.
YOU CAN ALSO THINK OF THE AMPLITUDE
AS HALF THE VERTICAL DISTANCE
FROM THE MAXIMUM AND MINIMUM VALUES.
SO LET'S TAKE A LOOK AT A COUPLE GRAPHS HERE.
HERE WE SEE THE GRAPH OF Y = SINE X.
THIS IS THE CENTER OF THE WAVE
AND YOU CAN SEE FROM THE CENTER TO THIS MAXIMUM = 1 UNIT.
THEREFORE THE AMPLITUDE = 1.
BUT IF WE LOOK AT THE NEXT GRAPH, Y = 3 SINE X,
TAKE A LOOK AT THE CENTER OF THE WAVE AGAIN.
FROM THE CENTER TO THE MAXIMUM IS 3 UNITS,
AS WELL AS FROM THE CENTER TO THE MINIMUM.
SO THE AMPLITUDE NOW = 3.
AGAIN, YOU CAN THINK OF THE AMPLITUDE
AS HALF THE DISTANCE FROM THE MAX TO THE MIN.
THE DISTANCE FROM -3 TO 3 IS 6 UNITS.
HALF OF 6, OF COURSE, IS 3. AND LASTLY, WE HAVE Y = -2 SINE X.
AGAIN, THE DISTANCE FROM THE CENTER TO THE MAX OR THE MIN
IS EQUAL TO TWO UNITS HERE, THEREFORE THE AMPLITUDE = 2.
AND AMPLITUDE ALWAYS HAS TO BE POSITIVE,
SO WE DO NOT WANT TO INCLUDE THIS NEGATIVE HERE.
REMEMBER THIS NEGATIVE
HAS THE AFFECT OF A REFLECTION ACROSS THE X-AXIS. OKAY.
SO IN GENERAL,
IF WE HAVE THE GRAPH OF Y = "A" SINE X OR Y = "A" COSINE X
THE AMPLITUDE IS THE ABSOLUTE VALUE "A".
REMEMBER IT ALWAYS HAS TO BE POSITIVE.
AND THE RANGE OF THE FUNCTION WILL BE FROM
NEGATIVE ABSOLUTE VALUE OF "A" TO "A" ON THE CLOSED INTERVAL.
CHANGING "A" RESULTS IN WHAT IS CALLED THE VERTICAL DILATION,
WHICH CAN ALSO BE THOUGHT OF AS THE VERTICAL STRETCH
OR COMPRESSION. OKAY.
LET'S PUT BOTH OF THESE IDEAS TOGETHER
BY STATING THE AMPLITUDE PERIOD AND THEN GRAPHING THE FUNCTION.
SO WE HAVE Y = SINE OF X DIVIDED BY 3.
SO FROM THIS WE CAN GATHER THAT "A" = 1 AND B = TO--
REMEMBER THERE'S A 1 HERE, SO 1/3.
SINCE "A" = 1 OR AMPLITUDE = 1, THE PERIOD IS 2PI DIVIDED BY B.
SINCE B IS 1/3,
2PI DIVIDED BY 1/3 IS THE SAME AS 2PI x 3/1 OR 6PI.
SO THE NEXT STEP IS WE MARK OFF ONE PERIOD ON OUR GRAPH
ON OUR X-AXIS FROM 0 TO 6PI.
THEN WE'LL DIVIDE THIS INTERVAL INTO FOUR EQUAL PARTS.
SO HALF OF 6PI WOULD BE 3PI,
AND THEN DIVIDING IN HALF AGAIN WE'D HAVE 3PI/2.
AND THEN HERE WE WOULD HAVE 9PI/2.
NOW, BEFORE WE BEGIN LETS REVIEW THE BASIC SINE FUNCTION HERE.
THIS ALSO HAS AN AMPLITUDE OF ONE,
BUT THE PERIOD HAS CHANGED FOR THE GIVEN FUNCTION.
BUT THAT'S REALLY THE ONLY DIFFERENCE.
NOTICE IF WE WERE TO TAKE THIS FUNCTION
AND DIVIDE IT INTO FOURTHS--
NOTICE IF WE WERE TO TAKE THIS PERIOD
AND DIVIDE IT INTO FOURTHS
THE FIRST FOURTH WOULD BE OUR MAXIMUM,
THE SECOND FOURTH WOULD BE BACK TO 0,
THE THIRD FOURTH WOULD BE AT -1,
AND THE FOURTH FOURTH WOULD BE BACK AT 0
FOR ONE COMPLETE CYCLE OF THE SINE FUNCTION.
SO THE ONLY DIFFERENCE HERE IS NOW OUR PERIOD IS 6PI,
SO WE'RE GOING TO START AT 0.
THEN AT 3PI/2 WE'LL BE UP AT 1 SINCE OUR AMPLITUDE IS STILL 1.
AT 3PI WE'LL BE BACK AT 0. 9 PI/2 WE'LL BE DOWN AT -1.
AND THEN AT 6PI WE'LL BE BACK 1.
AND NOW WE CAN MAKE A NICE GRAPH OF Y = SINE OF X DIVIDED BY 3.
NOTICE THE GRAPHS LOOK VERY SIMILAR.
THE ONLY DIFFERENCE IS WE HAVE A DIFFERENT PERIOD.
NOW LET'S GRAPH Y = 1/2 COSINE 2X.
NOTICE FOR REFERENCE WE'LL USE
THE BASIC GRAPH OF Y = COSINE THETA.
OKAY. SO FROM THIS WE CAN GATHER THAT "A" = 1/2, B = 2.
SINCE "A" = 1/2 OUR AMPLITUDE = 1/2,
AND OUR PERIOD = 2PI DIVIDED BY B.
SO WE HAVE A PERIOD OF PI RADIANS.
FIRST STEP WOULD BE TO MARK OFF PI RADIANS ON THE X AXIS,
DIVIDE IT INTO FOUR EQUAL PARTS.
SO HERE'S OUR PI AND WE HAVE PI/2.
THIS WOULD BE PI/4, AND THIS WOULD BE 3PI/4.
AGAIN, NOTICE THAT FOR 0 RADIANS THE BASIC COSINE FUNCTION = +1.
WELL, IF OUR AMPLITUDE IS NOW ONLY 1/2,
AT 0 RADIANS THE FUNCTION VALUE WILL BE 1/2 RATHER THAN 1.
AND NOW THE PATTERN WILL BE CONSISTENT.
AT THE NEXT FOURTH WILL BE DOWN AT 0.
AT THE NEXT FOURTH WILL BE DOWN AT -1,
BUT IN THIS CASE -1/2, BACK TO 0, AND THEN BACK UP TO 1/2.
THESE POINTS ARE ENOUGH TO MAKE A NICE GRAPH
OF Y = 1/2 COSINE 2X.
NOW, NOTICE THAT THESE FUNCTIONS LOOK ALMOST THE SAME.
WHAT WE'VE CHANGED IS HOW WE SCALED THE X-AXIS
AND HOW WE SCALED THE Y-AXIS. LET'S GO AND TRY ONE MORE.
NOW WE HAVE Y = -2 SINE 3X.
NOTICE, AGAIN, FOR REFERENCE WE HAVE Y = SINE THETA.
THIS SHOULD HELP US SEE THE PATTERN FOR THE NEW GRAPH.
REMEMBER THE AMPLITUDE = THE ABSOLUTE VALUE, SO IT'S = 2.
AND THE PERIOD = 2PI DIVIDED BY B.
SO OUR PERIOD = 2PI DIVIDED BY 3.
SO WE'LL START BY MARKING OFF OUR X-AXIS AT 2 PI/3.
SO HERE WE HAVE (0,2 PI/3) HERE.
WE'LL DIVIDE THIS INTERVAL INTO FOUR EQUAL PARTS.
SO IN THE MIDDLE WE HAVE PI/3, HALF OF THAT WOULD PI/6.
AND HALF THE DISTANCE BETWEEN PI/3 AND 2PI/3 WOULD BE PI/2.
OKAY. REMEMBER THAT AT 0 RADIANS THE BASIC SINE FUNCTION = 0,
SO THAT'LL STILL BE TRUE HERE.
BUT NOW ON THE BASIC SINE FUNCTION
TO THE FIRST FOURTH WE'RE UP AT +1.
SINCE THE AMPLITUDE = 2
WE MIGHT THINK WE'RE GOING TO BE UP AT +2,
BUT REMEMBER THAT THIS NEGATIVE HERE RESULTS
IN A REFLECTION ACROSS THE X-AXIS,
SO WE'RE ACTUALLY DOWN AT -2.
SO YOU DO HAVE TO BE CAREFUL, WHEN "A" IS NEGATIVE,
YOU DO HAVE TO REFLECT YOUR FUNCTION VALUES
ACROSS THE X-AXIS.
THE SECOND FOURTH WOULD BE BACK AT 0.
NOTICE ON THE THIRD FOURTH WE'RE DOWN AT A MINIMUM,
BUT SINCE WE'RE REFLECTING ACROSS THE X-AXIS HERE,
WE'LL BE AT A MAXIMUM.
IN THIS CASE, SINCE THE AMPLITUDE = 2
WE'LL BE UP AT 2, AND THEN BACK TO 0.
SO NOW WE HAVE ENOUGH POINTS TO MAKE A NICE GRAPH
OF Y = -2 SINE 3X.
OKAY. I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING AND HAVE A GOOD DAY.