- WELCOME TO A VIDEO ON HOW TO DETERMINE THE EQUATION
OF A SINE AND COSINE FUNCTION.
THE GOAL OF THE VIDEO IS TO WRITE THE EQUATION
OF A SINE AND COSINE FUNCTION GIVEN A GRAPH.
SO WE'VE BEEN WORKING WITH TRANSFORMATIONS
OF THE SINE AND COSINE FUNCTION, SO WE SHOULD BE FAMILIAR
WITH THESE TWO EQUATIONS AND WHAT THESE VARIABLES,
A, B, C, AND D TELL US ABOUT THE FUNCTION.
AND ONE OF THE ISSUES WITH FINDING THE EQUATION
OF A SINE AND COSINE GRAPH IS THAT IT'S NOT UNIQUE.
WE COULD USE EITHER SINE OR COSINE,
SO, REALLY, WE HAVE A CHOICE ON WHICH OF THESE EQUATIONS
WE WANT TO USE.
AND THE WAY WE'RE GOING TO DO THIS
IS FIRST WE'RE GOING TO PICK A PIECE OF THE GRAPH,
AND WE'RE GOING TO HIGHLIGHT ONE PERIOD OF IT.
SO FOR THIS GRAPH I'M GOING TO ISOLATE THIS PIECE OF THE GRAPH.
AND YOU CAN SEE RIGHT AWAY IT RESEMBLES THE GRAPH
OF SINE THETA.
SO WE'RE ACTUALLY GOING TO USE THIS FORM
TO FIND THE EQUATION OF THIS GRAPH.
THE NEXT THING I RECOMMEND IS DRAWING A HORIZONTAL LINE
THROUGH THE CENTER OF THE GRAPH LIKE THIS.
WHAT THIS DOES IS MAKE FINDING THE AMPLITUDE EASIER.
AND WE CAN SEE, THIS IS A CENTER LINE
AND THIS IS THE MAXIMUM.
THIS DISTANCE IS 1/2, THEREFORE AMPLITUDE
IS EQUAL TO 1/2. AND SO WE KNOW THAT A = 1/2.
AND THE NEXT THING IT SHOWS US IS THAT TYPICALLY
THE SINE FUNCTION HUGS THE X-AXIS WHERE THE X-AXIS
IS THE CENTER OF THE GRAPH.
AND WE CAN SEE NOW IT'S BEEN SHIFTED UP ONE,
SO NOW WE KNOW THAT C IS EQUAL TO 1.
SO NOW WHAT WE HAVE TO DO IS FIND THE PERIOD
AND THE PHASE SHIFT OR HORIZONTAL SHIFT.
WHAT WE CAN ALSO TELL THAT FROM THE BASIC SINE FUNCTION,
THIS GRAPH HAS BEEN SHIFTED PI OVER 4 UNITS TO THE RIGHT.
SO THAT TELLS US THAT D IS GOING TO BE PI OVER 4.
AND LASTLY, WE NEED TO FIND THE VALUE OF B.
AND WE CAN FIGURE THAT OUT FROM THE PERIOD.
NOTICE THAT FROM PI OVER 4 TO 5PI OVER 4,
WE HAVE ONE COMPLETE CYCLE OF THIS SINE FUNCTION.
SO 5PI OVER 4 - 1PI OVER 4 IS 4PI OVER 4, OR PI RADIANS.
SO THE PERIOD IS PI RADIANS.
REMEMBER THE PERIOD IS EQUAL TO 2PI DIVIDED BY B.
AND, AGAIN, WE'RE SAYING THAT'S EQUAL TO PI RADIANS.
SO WE CAN DO CROSS PRODUCTS HERE.
WE HAVE B PI IS EQUAL TO 2PI.
DIVIDING BY PI, WE HAVE B = 2.
THIS IS ALL THE INFORMATION WE NEED.
WE HAVE A, B, C, AND D.
SO LET'S WRITE OUR EQUATION.
Y = 1/2 SINE
OF THE QUANTITY B
TIMES THE QUANTITY (X - D).
D IS PI OVER 4 + C. AND OUR C VALUE IS 1.
OKAY, SO LET'S CHECK THIS OUT. OUR AMPLITUDE IS 1/2.
OUR PERIOD, 2PI DIVIDED BY 2 IS PI RADIANS.
THE PHASE SHIFT, REMEMBER, IF IT'S -PI OVER 4
THAT MEANS RIGHT PI OVER 4 AND THEN UP 1.
SO, THERE WE GO. LET'S TRY ANOTHER.
FIRST, LET'S TAKE A LOOK AT THE GRAPH AND DETERMINE
WHICH PIECE OF THE GRAPH THEY WANT TO FOCUS ON.
WANT TO HIGHLIGHT ONE PERIOD OF WHAT LOOKS LIKE
EITHER THE GRAPH OF SINE OR COSINE.
AND THIS IS WHY THE EQUATION THAT WE FIND WILL NOT BE UNIQUE.
SO WHEN I LOOK AT THIS, I SEE THE GRAPH OF A SINE FUNCTION
STARTING AT PI OVER 4, GOING UP, DOWN, AND THEN BACK UP.
SO WE'LL USE THIS PIECE TO FIND THE EQUATION
OF THIS FUNCTION.
AND, AGAIN, YOU CAN SEE WE'RE GOING TO USE THE SINE FUNCTION.
NEXT, WE'LL DRAW A LINE THROUGH THE CENTER.
LOOKS LIKE IT'S GOING TO BE AT -1.
SO, RIGHT AWAY WE CAN DRAW THE CENTER LINE THROUGH Y = -1.
WE KNOW THAT OUR VERTICAL SHIFT C WILL BE -1.
NEXT, THE AMPLITUDE FROM -1 TO +2 WOULD BE THREE UNITS.
SO A = 3.
THE PHASE SHIFT, AGAIN, IS RIGHT PI OVER 4 UNITS.
SO D = PI OVER 4.
AND NOW WE HAVE TO DETERMINE THE PERIOD
SO WE CAN FIND THE VALUE OF B.
REMEMBER, 2PI DIVIDED BY B IS EQUAL TO OUR PERIOD.
SO WE HAVE 3PI OVER 4 - 1PI OVER 4
WOULD BE 2PI OVER 4 OR PI OVER 2.
AND, AGAIN, WE'LL PERFORM CROSS PRODUCTS HERE.
B x PI MUST EQUAL 4PI.
DIVIDING BOTH SIDES BY PI, WE HAVE B = 4.
AND THAT'S ALL WE NEED.
AGAIN, WE'RE FOCUSING ON THE SINE FUNCTION.
SO WE HAVE Y = 3 SINE OF THE QUANTITY B,
WHICH IS 4, (X - D), SO X - PI OVER 4.
AND THE SHIFT WAS DOWN 1, SO C = -1.
OKAY, LET'S TAKE A LOOK AT ONE MORE.
LET'S IDENTIFY THE PIECE WE WANT TO FOCUS ON.
AND FOR THIS ONE I'M GOING TO FOCUS ON COSINE.
SO WE'LL START HERE AT THIS POINT
AND END AT THIS POINT.
BECAUSE FROM THERE ON IT STARTS TO REPEAT.
DRAW A LINE THROUGH THE CENTER.
LOOKS LIKE IT'S AT Y = -2.
SO RIGHT AWAY WE KNOW THAT C = -2.
NEXT, WE CAN SEE THE DISTANCE FROM OUR CENTER
TO A MAXIMUM IS TWO UNITS.
WE'RE LOOKING AT THIS GRAPH ON THE INTERVAL FROM -PI OVER 4
TO 7PI OVER 4.
THAT HORIZONTAL DISTANCE IS 2PI.
SO OUR PERIOD IS 2PI, WHICH MAKES B = 1.
AND LASTLY, WE NEED TO FIND D.
THE HORIZONTAL SHIFT OR PHASE SHIFT
IS LEFT PI OVER 4 UNITS, SO D IS ACTUALLY -PI OVER 4.
NOW, THERE'S ONE OTHER THING.
TYPICALLY WHEN WE GRAPH THE COSINE FUNCTION
ON THE INTERVAL FROM 0 TO 2PI, IT LOOKS SOMETHING LIKE THIS.
AND NOTICE THAT WE USUALLY START AT A MAXIMUM.
BUT NOW WE'RE STARTING AT A MINIMUM.
SO WHAT HAPPENED WAS THE COSINE FUNCTIONING
WAS REFLECTED ACROSS THE X-AXIS BEFORE IT WAS SHIFTED DOWN
TWO UNITS.
THE RESULT IS "A" IS EQUAL TO -2.
SO WE HAVE ALL THE INFORMATION WE NEED NOW.
WE HAVE Y = -2 COSINE, OUR B VALUE IS 1,
SO WE'LL LEAVE THAT OFF.
AND THEN WE HAVE X - D,
OR MINUS A -PI OVER 4 IS PLUS PI OVER 4,
AND THEN MINUS 2.
NOW, IN THIS PROBLEM, I DID PICK A MORE CHALLENGING
PIECE OF THE GRAPH TO FIND THE EQUATION TO IT TIME.
REMEMBER, THESE EQUATIONS ARE NOT UNIQUE,
SO WE COULD HAVE CHOSEN A DIFFERENT PIECE OF THIS GRAPH
TO FIND AN EQUIVALENT EQUATION IN A DIFFERENT FORM.
OKAY. I HOPE YOU FOUND THIS VIDEO HELPFUL. THANK YOU.