Angles in Standard Position

Angles in Standard Position


– WELCOME TO A PRESENTATION
ON ANGLES IN STANDARD POSITION. THE GOALS OF THIS VIDEO
ARE TO PLOT ANGLES IN STANDARD POSITION, AND ALSO TO DETERMINE
COTERMINAL ANGLES. AN ANGLE IS IN STANDARD POSITION IF ITS VERTEX IS AT THE ORIGIN
AND THE INITIAL SIDE IS ON THE POSITIVE X-AXIS. SO HERE WE HAVE TWO ANGLES
IN STANDARD POSITION BECAUSE THE VERTEX
IS AT THE ORIGIN, AND THE INITIAL SIDE
OF THE ANGLE IS ALONG THE POSITIVE X-AXIS. AN ANGLE IN STANDARD POSITION
IS SAID TO LIE IN THE QUADRANT IN WHICH ITS TERMINAL SIDE LIES. SO IF WE THOUGHT OF THIS ANGLE
AS THE ROTATION FROM THIS INITIAL SIDE
TO THIS TERMINAL SIDE, IT WOULD LIE
IN THE FIRST QUADRANT. AND WE SHOULD ALSO NOTE THERE’S AN INFINITE NUMBER
OF ANGLES THAT WOULD ALSO HAVE THIS
AS ITS TERMINAL SIDE. FOR EXAMPLE, IF WE STARTED
HERE AT THE INITIAL SIDE AND ROTATED IT CLOCKWISE,
THIS WOULD BE A NEGATIVE ANGLE, WHICH WOULD ALSO LIE
IN THE FIRST QUADRANT. AND THE SAME THING
WITH THE SECOND ANGLE. IF WE CONSIDER THE ANGLE WHERE IT’S ROTATED
FROM THIS INITIAL SIDE TO THIS TERMINAL SIDE, THIS ANGLE WOULD LIE
IN THE SECOND QUADRANT. BUT, AGAIN, SO WOULD THIS
NEGATIVE ANGLE HERE. SO IN GENERAL, ANGLES BETWEEN
ZERO AND 90 DEGREES LIE IN THE FIRST QUADRANT, BETWEEN 90 AND 180,
SECOND QUADRANT, BETWEEN 180 AND 270
THE THIRD QUADRANT, AND BETWEEN 270 AND 360
IN THE FOURTH QUADRANT. NOW, IF THE TERMINAL SIDE
OF AN ANGLE HAPPENS TO LIE ON ONE OF THE AXES,
IT’S CALLED A QUADRANTAL ANGLE. SO ANGLES LIKE 90, 180, 270,
AND SO-ON WOULD BE QUADRANTAL ANGLES. LET’S GO AHEAD AND TALK
ABOUT COTERMINAL ANGLES NOW. COTERMINAL ANGLES ARE ANGLES
IN STANDARD POSITION THAT HAVE A COMMON
TERMINAL SIDE. SO, FOR EXAMPLE, 30 DEGREES,
390 DEGREES, AND -330 DEGREES
ARE ALL COTERMINAL ANGLES. AND LET’S TAKE A LOOK AT WHY. LET’S PLOT EACH OF THESE. HERE’S OUR INITIAL SIDE. LET’S GO AHEAD
AND PLOT 30 DEGREES NOW. SO ROTATE COUNTERCLOCKWISE
30 DEGREES, WHICH REMEMBER WOULD BE
ABOUT A THIRD OF THE ROTATION FROM ZERO TO 90 DEGREES. NOW, IF WE TRY
TO PLOT 390 DEGREES, REMEMBER ONE COMPLETE REVOLUTION
IS 360 DEGREES. SO IF I START HERE, ROTATE COUNTERCLOCKWISE
ONE COMPLETE REVOLUTION THAT WOULD BE 360 DEGREES. AND TO PLOT 390,
I NEED TO GO 30 MORE DEGREES. SO, AS YOU CAN SEE, IF I ROTATE
30 MORE DEGREES FROM HERE IT’S GOING TO BE COTERMINAL
WITH +30 DEGREES BECAUSE THE TERMINAL SIDE
OF THE ANGLE ENDS UP IN THE SAME POSITION. AND LASTLY,
-330 DEGREES WOULD START HERE AND THEN GO
CLOCKWISE 330 DEGREES, WHICH, AGAIN, WOULD BE
30 DEGREES SHY OF ONE COMPLETE
REVOLUTION CLOCKWISE. AND, AGAIN, THE TERMINAL SIDE
OF THE ANGLE ENDS UP IN THE SAME POSITION
FOR ALL THREE OF THESE. THEREFORE, THEY ARE CALLED
COTERMINAL ANGLES. LET’S FIND TWO ANGLES THAT ARE
COTERMINAL WITH 135 DEGREES. SO LET’S FIRST PLOT 135 DEGREES
IN STANDARD POSITION. WE’LL ROTATE COUNTERCLOCKWISE
135 DEGREES. BUT FROM HERE TO HERE
WE’D BE AT 90 AND WE NEED TO ROTATE
45 MORE DEGREES. SO HERE’S OUR GIVEN ANGLE,
135 DEGREES. SO I NEED TO FIND
TWO OTHER ANGLES THAT HAVE A TERMINAL SIDE
IN THIS SAME POSITION. WELL, REMEMBER THAT 360 DEGREES
ANOTHER REVOLUTION. SO IF WE TAKE 135 DEGREES
AND ADD 360 DEGREES TO THAT, THAT WOULD BE JUST ONE MORE
REVOLUTION PAST THIS ANGLE, WHICH WOULD BE 495 DEGREES. SO WE START HERE,
ROTATE ONE COMPLETE REVOLUTION, THERE’S 360,
AND THEN ANOTHER 135 DEGREES, WHICH WOULD END UP HERE BEING COTERMINAL
WITH THE INITIAL ANGLE. NOW, FOR THE LAST ANGLE, LET’S GO AHEAD AND FIND
A NEGATIVE COTERMINAL ANGLE. SO WHAT WE WANT TO DO, IS STARTING
WITH THIS INITIAL SIDE, WE WANT TO ROTATE CLOCKWISE
THIS MANY DEGREES. SO WE WANT TO START HERE, AND ROTATE CLOCKWISE
OVER TO HERE. SO THE QUESTION
IS WHAT ANGLE WOULD THAT BE. BUT IF WE LOOK AT IT LOGICALLY,
THIS WOULD BE -90, -180, AND THEN -45 DEGREES MORE. THAT WOULD BE -225 DEGREES. OR WE COULD ALSO START
WITH 135 DEGREES AND SUBTRACT 360 DEGREES, WHICH REPRESENTS
A REVOLUTION CLOCKWISE. SO WE’D START HERE
AND GO CLOCKWISE 360 DEGREES, AND WE GET THE SAME THING
OF -225 DEGREES. LET’S TAKE A LOOK
AT A COUPLE MORE EXAMPLES. WE WANT TO FIND ANGLES
OF LEAST POSITIVE MEASURE THAT ARE COTERMINAL
TO EACH ANGLE. SO, AGAIN, THERE’S A COUPLE WAYS
OF DOING THIS. LET’S FIRST SKETCH
OUR INITIAL SIDE. AND WE’RE GOING TO START
ROTATING COUNTERCLOCKWISE MULTIPLES OF 360 DEGREES. SO IF WE START HERE
AND ROTATE ONCE COUNTERCLOCKWISE THAT WOULD BE 360 DEGREES. AND WE CAN DO THAT AGAIN, THAT WOULD BE
ANOTHER 360 DEGREES, WHICH BRINGS US TO 720 DEGREES. AGAIN, WE’RE TRYING
TO REACH 1,070. SO LET’S TRY
ONE MORE REVOLUTION. AND WHEN WE DO THAT,
WE END UP AT 1,080 DEGREES, WHICH IS 10 DEGREES TOO FAR. SO WE ACTUALLY
HAVE TO STOP 10 DEGREES SHY OF ANOTHER REVOLUTION. THIS IS THE GIVEN ANGLE
OF 1,070 DEGREES. SO IF WE WANTED THE LEAST
POSITIVE COTERMINAL ANGLE IT WOULD BE 10 DEGREES SHY OF
ONE REVOLUTION OR 350 DEGREES. NOW, ANOTHER WAY
TO OBTAIN 350 DEGREES WOULD BE TO START
WITH THE INITIAL ANGLE AND FIGURE OUT HOW MANY
REVOLUTIONS ARE IN THIS. SO IF WE DO THIS,
360 GOES INTO 1070 TWO TIMES WITH THE REMAINDER
OF 350 DEGREES. AND THIS REMAINDER
DOES REPRESENT OUR POSITIVE COTERMINAL ANGLE. OKAY, LET’S TAKE A LOOK
AT PART B, -65 DEGREES. I ALWAYS LIKE TO START
BY SKETCHING THE GIVEN ANGLE. THERE’S OUR INITIAL SIDE. THIS TIME WE’LL GO
CLOCKWISE 65 DEGREES, SO OUR TERMINAL SIDE WOULD BE
SOMEWHERE IN HERE.   AND THE LEAST POSITIVE
COTERMINAL ANGLE WOULD BE THIS ANGLE HERE. WHAT WE CAN DO, IS START WITH THE INITIAL ANGLE
OF -65 DEGREES AND ADD ANOTHER REVOLUTION
COUNTERCLOCKWISE, WHICH WOULD GIVE US 295 DEGREES, WHICH IS THE LEAST POSITIVE
COTERMINAL ANGLE TO -65. OKAY. LET’S TAKE A LOOK
AT ONE MORE QUESTION. WHAT WOULD BE AN EXPRESSION YOU COULD USE TO EXPRESS ALL
ANGLES COTERMINAL TO 90 DEGREES? SO HERE’S THE GIVEN ANGLE. WE WANT ALL THE ANGLES
THAT ARE COTERMINAL TO THIS, WHICH MEANS FROM HERE WE’D HAVE
TO EITHER GO AROUND CLOCKWISE OR COUNTERCLOCKWISE
FROM THIS ANGLE. WHICH REMEMBER ONE REVOLUTION
REPRESENTS 360 DEGREES. SO WE WOULD START
WITH 90 DEGREES AND WE COULD ADD MULTIPLES
OF 360 DEGREES. SO IF WE MULTIPLY 360
BY SUM INTEGER N, THAT WOULD GIVE US
ANY COTERMINAL ANGLE TO 90 DEGREES,
WHERE N IS SUM INTEGER. OKAY. I HOPE YOU FOUND
THIS VIDEO HELPFUL. THANK YOU FOR WATCHING.  

7 Comments

Leave a Reply

Your email address will not be published. Required fields are marked *