Graphing the Cotangent Function


[ MUSIC ] – WELCOME TO A LESSON
ON THE GRAPH OF THE COTANGENT FUNCTION. THE GOALS OF THE VIDEO ARE TO DETERMINE THE GRAPH
OF COTANGENT AND ALSO TO DETERMINE
THE KEY PROPERTIES OF THE COTANGENT GRAPH. SO SINCE COTANGENT AND TANGENT
ARE RECIPROCAL FUNCTIONS, IF WE DRAW A TANGENT LINE
TO A POINT ON THE Y AXIS, RATHER THAN THE X AXIS, WE CAN FORM A COTANGENT
SEGMENT HERE IN RED. AND THE LENGTH OF THIS SEGMENT
IS EQUAL TO COTANGENT THETA. AND NOTICE THAT THIS IS
ANGLE THETA HERE IN STANDARD POSITION, THIS IS ALSO ANGLE THETA. SO IN A SIMILAR FASHION
TO OUR TANGENT SEGMENT, WE CAN USE SIMILAR TRIANGLES TO SHOW THAT COTANGENT THETA
IS EQUAL TO T, THIS BEING
THE COTANGENT SEGMENT. REMEMBER THAT WHEN WE HAVE
AN ANGLE IN STANDARD POSITION, COTANGENT THETA
WOULD BE EQUAL TO X/Y. SO WE WANT TO SHOW THAT T
IS EQUAL TO X/Y. WELL BECAUSE THESE TWO
TRIANGLES ARE SIMILAR DUE TO ANGLE,
ANGLE SIMILARITY, WE CAN SET UP A PROPORTION
USING THE CORRESPONDING SIDES SUCH THAT X/Y IS EQUAL TO T/1 AND SINCE THEY’RE SIMILAR
TRIANGLES, AND THIS IS UPPER PORTION, — PRODUCTS MUST BE EQUAL. SO WE CAN MULTIPLY X x 1 TO
THE X AND T x Y TO GET T, Y. THEN TO SOLVE FOR T
WE CAN DIVIDE BOTH SIDES BY Y AND WE HAVE NOW SHOWN
THAT T IS EQUAL TO X/Y. SO THE COTANGENT SEGMENT
IS EQUAL TO THE RATIO OF X/Y. LET’S GO AHEAD AND LOOK
AT A WOLFRAM DEMONSTRATION TO ILLUSTRATE THIS. SO AGAIN IN RED WE HAVE
THE COTANGENT SEGMENT SO AT AN ANGLE OF 0 RADIANS,
OR 0 DEGREES, THE COTANGENT SEGMENT
HAS AN INFINITE LENGTH AND THEREFORE THERE’S GOING
TO BE A VERTICAL ASYMPTOTE AT 0 DEGREES OR 0 RADIANS. NOTICE AS WE APPROACH
90 DEGREES OR PI/2 RADIANS, WE HAVE VERY LONG COTANGENT
SEGMENTS WRAPPED HERE
ON THE COORDINATE PLANE BUT AS THE ANGLE APPROACHES,
PI/2 RADIANS, IT’S SHRINKING AND AT PI/2 WILL HAVE THE
LENGTH OF 0 AS WE SEE HERE. SO HERE’S ONE PIECE
OF THE COTANGENT FUNCTION. THEN AS WE PROGRESS
TOWARD PI RADIANS, AGAIN THE SEGMENTS
ARE VERY SHORT BUT BECAUSE THIS SEGMENT
IS LEFT OF THE Y AXIS, WE’RE GIVING IT
A NEGATIVE VALUE AND AS WE APPROACH PI RADIANS IT’S GETTING LONGER AND LONGER
IN THE NEGATIVE DIRECTION AND AGAIN YOU CAN SEE IT’S
APPROACHING NEGATIVE INFINITY. SO AGAIN WITH PI
WE HAVE A VERTICAL ASYMPTOTES. AS WE CONTINUE
AROUND THE UNIT CIRCLE, AGAIN YOU CAN SEE
THE COTANGENT SEGMENT IN RED. IT’S APPROACHING 0 AS WE
APPROACH 3 PI OVER 2 RADIANS AND THEN AS WE APPROACH
2 PI RADIANS WHICH IS NEGATIVE APPROACHING NEGATIVE INFINITY. AND SO FROM ONE REVOLUTION
AROUND THE UNIT CIRCLE, WE’VE ACTUALLY COMPLETED
TWO COMPLETE CYCLES OF THE COTANGENT FUNCTION AND WE CAN SEE THE PERIOD
WOULD BE PI RADIANS. SO TO GRAPH THIS BY HAND WE’RE
ACTUALLY GOING TO USE THE FACT THAT COTANGENT AND TANGENT
ARE RECIPROCAL FUNCTIONS TO PRODUCE THE GRAPH. SO WE’LL ASSUME IT CAN GRAFT
THE TANGENT FUNCTION AND FROM THAT WE’LL GRAPH
THE COTANGENT FUNCTION BY TAKING RECIPROCAL VALUES
OF TANGENT THETA. SO IN GRAY WE HAVE TANGENT
THETA ALREADY GRAPHED FOR US SO WE’RE GOING TO PICK
SOME KEY VALUES ON THE TANGENT FUNCTION, FIND THE RECIPROCAL VALUES
AND PLOT THOSE FOR COTANGENT. AND WE’LL START OFF
BY FIGURING OUT THAT WHERE TANGENT
IS EQUAL TO 0, RECIPROCAL OF 0
WOULD BE UNDEFINED, SO THE RESULT WOULD BE
VERTICAL ASYMPTOTES AT 0, PI AND 2 PI. NEXT WE KNOW THE RECIPROCAL
OF 1 IS 1, RECIPROCAL OF -1 IS STILL -1 AND RECIPROCAL OF AS WELL
TO THE RIGHT. NEXT FOR THE SAME REASON
WHY WHEN TANGENT IS 0 COTANGENT IS UNDEFINED, WHERE TANGENT IS UNDEFINED, WHERE THERE’S A VERTICAL
ASYMPTOTE COTANGENT WOULD BE 0. AND LET’S PICK A FEW MORE
KEY VALUES WHERE TANGENT
IS EQUAL TO .5 OR 1/2, COTANGENT WOULD BE 2
HERE, HERE AND HERE. SO WE ALREADY HAVE THESE
VERTICAL ASYMPTOTES HERE AND WE HAVE THESE FOUR POINTS
HERE. LET’S GO AHEAD AND TRY TO FIND
A COUPLE MORE KEY VALUES. HERE WHERE TANGENT THETA
IS EQUAL TO -1/2, TYPICAL OF -1/2 IS -2. HERE AND HERE AND LASTLY, LET’S SAY THIS IS WHERE
TANGENT IS EQUAL TO 1/4, RECIPROCAL OF 1/4 WOULD BE 4. AND HERE WE’LL APPROXIMATE
THE VALUE OF -1/4, THE RECIPROCAL WOULD BE -4. AND VERY QUICKLY WE CAN SEE WE HAVE A NICE GRAPH
OF Y=COTANGENT THETA. AND SINCE THE PERIOD
OF COTANGENT THETA IS EQUAL TO PI RADIANS, WE’LL HAVE A SIMILAR GRAPH BETWEEN THESE TWO VERTICAL
ASYMPTOTES AND SO ON. NOW OF COURSE IF THIS WAS
HARD TO FOLLOW, YOU CAN ALWAYS PULL UP
YOUR GRAPHING CALCULATOR AND FIND FUNCTION VALUES
THAT WAY. SO MY CALCULATOR’S
IN DEGREE MODE AND FOR Y WHEN I TYPED 1
OVER TANGENT X, SINCE THERE’S NO COTANGENT KEY
AND THEN JUST TO COMPARE, IN Y2 I ENTER TANGENT X. SO Y1 WILL BE OUR COTANGENT
FUNCTION AND Y2 WILL BE
THE TANGENT FUNCTION. SO I’M GOING TO GO AHEAD
AND PRESS SECOND GRAPH AND AGAIN NOTICE
THAT AT 0 DEGREES THERE’S AN ERROR FOR COTANGENT MEANING THERE’S
A VERTICAL ASYMPTOTE. TANGENT WAS 0, AGAIN AT 15 DEGREES
WE HAVE APPROXIMATELY 4 OR SOMEWHERE IN HERE AT
45 DEGREES PI OVER 4 RADIANS WE’RE AT 1. NOW NOTICE THE CALCULATOR
DOES HAVE A PROBLEM AT 90 DEGREES
OR PI OVER 2 RADIANS BECAUSE WE’RE USING
THE TANGENT FUNCTION TO EVALUATE THIS AND SINCE AT 90 DEGREES
TANGENT IS UNDEFINED, THE CALCULATOR CANNOT EVALUATE
THIS FOR COTANGENT THETA. SO IT IS IMPORTANT THAT WE
RECOGNIZE THE RELATIONSHIP BETWEEN THE GRAPH OF TANGENT
THETA AND COTANGENT THETA. BUT AGAIN THIS CAN BE HELPFUL FOR MOST VALUES
OF COTANGENT THETA. AND LET’S GO AHEAD
AND TAKE A LOOK AT THE GRAPH
ON THE CALCULATOR. AGAIN I’M IN DEGREES SO I’M
GOING TO ENTER IN A WINDOW THAT’S APPROPRIATE FOR DEGREES
SO LET’S SAY -45, 405, X SCALE WILL GO BY 30 DEGREE
INCREMENTS AND THEN Y MINIMUM. LET’S JUST GO TO, LET’S GO TO
-5 TO 5 BY 1’S AND PRESS GRAPH AND THERE’S THE GRAPH
OF OUR COTANGENT FUNCTION. IT’S ALSO GOING TO GRAPH
THE TANGENT FUNCTION NOW AS WE SEE HERE. OKAY, LET’S SUMMARIZE THE KEY
COMPONENTS OF THE GRAPH OF Y=COTANGENT THETA. SO THE DOMAIN WOULD BE
ANY ANGLES — IF THETA IS NOT EQUAL
TO A MULTIPLE OF PI RADIANS AND AGAIN THAT’S BECAUSE THAT’S WHERE WE HAVE
OUR VERTICAL ASYMPTOTES. THE RANGE WOULD BE
FROM NEGATIVE INFINITY TO POSITIVE INFINITY. THE X INTERCEPTS OCCUR
AT ODD MULTIPLES OF PI OVER 2 SO 1 PI OVER 2, 3 PI OVER 2
AND SO ON. THE FUNCTION
HAS VERTICAL ASYMPTOTES. THE PERIOD IS PI RADIANS. THERE IS NO AMPLITUDE AND THE GRAPH IS SYMMETRICAL
WITH RESPECTS TO THE ORIGIN AND THEREFORE
IS AN ODD FUNCTION. OKAY. I HOPE YOU FOUND
THIS OVERVIEW HELPFUL. THANK YOU AND HAVE A GOOD DAY. [ MUSIC ]  

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