Prompt Title: Mathematics Transcript Study Guide

Summarize and write notes of the following; write questions for studying, and then answer those questions Title: "(15) Angles in Standard Position - YouTube" Transcript: "- 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."