- WELCOME TO A LESSON ON RADIAN MEASURE.
THE GOALS OF THIS VIDEO ARE TO DEFINE RADIAN MEASURE,
CONVERT ANGLES FROM DEGREES TO RADIANS
AND FROM RADIANS TO DEGREES,
AND ALSO TO DETERMINE TRIGONOMETRIC FUNCTION VALUES
GIVEN IN RADIANS.
JUST AS IT SOUNDS A RADIAN IS BASED ON THE RADIUS OF A CIRCLE.
ONE RADIAN IS THE ANGLE CREATED BY BENDING THE RADIUS LENGTH
AROUND THE ARC OF A CIRCLE.
AS WE SEE HERE THE RADIUS IN RED IS THE SAME LENGTH
AS THIS RED ARC AND, THEREFORE,
THIS IS AN ANGLE OF MEASURE ONE RADIAN.
SO BECAUSE A RADIAN IS BASED ON THE ACTUAL PART OF THE CIRCLE
RATHER THAN AN ARBITRARY DIVISION
IT IS A MUCH MORE NATURAL UNIT.
NOW, REMEMBER THAT THE CIRCUMFERENCE
OF A CIRCLE IS 2PI RADIANS.
SO IT FOLLOWS THAT 2PI RADIANS IS EQUAL TO 360 DEGREES.
SO IF WE TOOK THIS EQUATION
AND DIVIDED BOTH SIDES BY 360 DEGREES
WE COULD SHOW THAT PI RADIANS/180 DEGREES = 1.
IF WE DIVIDED BOTH SIDES BY 2PI RADIANS
WE COULD SHOW THAT 180 DEGREES DIVIDED BY PI RADIANS
IS EQUAL TO 1 AS WELL.
AND THESE ARE THE CONVERSIONS
THAT WE WILL USE TO CONVERT RADIANS TO DEGREES
AND VICE VERSA IN JUST A COUPLE MINUTES.
HOWEVER, LET'S FIRST TAKE A LOOK AT A WOLFRAM DEMONSTRATION.
WHAT WE SEE HERE IS A 30 DEGREE ANGLE IN STANDARD POSITION.
AND IF WE DIVIDE 30 DEGREES BY 360 DEGREES
WE WOULD SEE THAT THAT IS 1/12 OF THE WHOLE CIRCLE
OR 1/12 OF A FULL REVOLUTION.
SO, THEREFORE, IF WE WANTED TO CONVERT 30 DEGREES TO RADIANS
WE CAN MULTIPLY 1/12 BY 2PI RADIANS TO GET PI/6 RADIANS.
SO IN THIS ANIMATION AS WE CHANGE THE ANGLE
AT THE SAME TIME WE SEE THE CONVERSION TO RADIANS.
WE'LL STOP HERE AT 90 DEGREES
AND WE CAN SEE THAT 90 DEGREES IS 1/4 OF A REVOLUTION,
THEREFORE, 1/4 x 2PI WILL GIVE US PI/2 RADIANS.
AND YOU CAN SEE THIS ALSO MARKS OFF THE NUMBER OF RADIANS
ON THE OUTSIDE.
AGAIN, THIS JUST SHOWS THE RELATIONSHIP
BETWEEN ANGLES AND RADIANS.
AND, AGAIN, WE SEE 360 DEGREES IS EQUAL TO 2PI RADIANS.
AND A SIMILAR RELATIONSHIP OCCURS
WHEN WE HAVE NEGATIVE ANGLES.
LET'S GO AHEAD AND CONVERT SOME ANGLES AND DEGREES TO RADIANS.
AND WHAT WE DO IS WE'RE GOING TO MULTIPLY THE DEGREE
MEASURE BY PI/180 DEGREES AND THEN SIMPLIFY.
WELL, OF COURSE, SOME OF THESE WE SHOULD ALREADY KNOW,
MEANING, BY DEFINITION 360 DEGREES IS EQUAL TO 2PI RADIANS.
AND, OF COURSE, 180 DEGREES IS HALF OF 360 DEGREES,
SO HALF OF 2PI WOULD BE PI RADIANS.
LET'S GO AHEAD AND SHOW THIS ONE BY HAND.
IT'S HELPFUL TO PUT 90 DEGREES/1
AND THEN MULTIPLY IT BY PI/180 DEGREES.
AND WE NEED TO SIMPLIFY THIS.
THIS SIMPLIFIES NICELY.
THIS SIMPLIFIES TO 1.
THIS SIMPLIFIES TO 2.
NOTICE THE UNITS OF DEGREES SIMPLIFY OUT
AND WE'RE LEFT WITH PI/2 RADIANS.
SIMILARLY, 45 DEGREES IS HALF OF 90 DEGREES,
SO HALF OF PI/2 WOULD BE PI/4 RADIANS.
AND I'LL GO AHEAD AND SHOW THIS ONE AS WELL.
30 DEGREES/1 x PI/180 DEGREES, SIMPLIFYING.
30 DEGREES = PI/6 RADIANS,
AND THAT'S IMPORTANT BECAUSE A LOT OF THE ANGLES
THAT WE WILL ENCOUNTER WILL BE MULTIPLES OF 30 DEGREES
AND, THEREFORE, MULTIPLES OF PI/6.
AND 60 DEGREES IS 2 x 30 DEGREES.
SO WE COULD JUST TAKE PI/6 AND MULTIPLY BY 2
OR DOUBLE IT AND GET 2PI/6, WHICH IS PI/3 RADIANS.
BUT IF WE'RE EVER NOT SURE OF THE CONVERSION
WE CAN ALWAYS JUST MULTIPLY IT BY PI/180 DEGREES.
LET'S DO THE OPPOSITE NOW.
WE'LL CONVERT FROM RADIANS TO DEGREES.
AND NOTICE OFTEN WHEN AN ANGLE IS GIVEN IN RADIANS
THE UNITS ARE NOT GIVEN.
IT'S ASSUMED THEY ARE RADIANS.
AND, AGAIN, 2PI RADIANS = 360 DEGREES, THAT'S A GIVEN.
HALF OF 2PI IS PI, SO HALF OF 360 WOULD BE 180 DEGREES.
AND LET'S GO AHEAD AND SHOW THIS ONE.
WE WANT TO CONVERT PI/2 RADIANS TO DEGREES.
SO WE'LL MULTIPLY BY THIS TIME 180 DEGREES DIVIDED BY PI
AND SIMPLIFY.
PI SIMPLIFY OUT.
THIS SIMPLIFIES TO 90 DEGREES.
AND, AGAIN, THE RELATIONSHIP HERE, PI/4 IS HALF OF PI/2,
SO HALF OF 90 DEGREES WOULD BE 45 DEGREES.
AND, AGAIN, THIS IS IMPORTANT
BECAUSE IF WE HAVE AN ANGLE THAT'S A MULTIPLE OF 45 DEGREES,
WE WILL HAVE AN ANGLE THAT IS JUST A MULTIPLE OF PI/4 RADIANS.
PI/6 x 180 DEGREES/PI.
AGAIN, THE PI SIMPLIFY OUT. THERE ARE 30 SIXES IN 180,
SO THIS WOULD BE 30 DEGREES.
AND PI/12, THE PIs ARE OUT.
180 IS THE SAME AS 12 x 15, SO THIS WOULD CHANGE TO A 1
AND THIS CHANGES TO A 15 AND WE HAVE 15 DEGREES.
SO WHEN YOU'RE FIRST LEARNING TO WORK WITH RADIANS
ONE THING THAT CAN BE VERY HELPFUL IS TO REMEMBER
TWO KEY ANGLES IN RADIAN MEASURE
AND THE FIRST ONE IS 30 DEGREES IS EQUAL TO PI/6.
THE REASON THAT'S IMPORTANT IS IF WE HAVE AN ANGLE
THAT'S A MULTIPLE OF 30 DEGREES
IT'S GOING TO BE A MULTIPLE OF PI/6.
SO 60 DEGREES IS 2 x 30 DEGREES.
SO IT WOULD BE 2 x PI/6 OR 2PI/6, WHICH SIMPLIFIES TO PI/3.
90 DEGREES IS 3 x 30 DEGREES.
SO IT WOULD BE 3 x PI/6 OR 3PI/6,
WHICH SIMPLIFIES TO PI/2.
120 DEGREES IS 4 x 30 DEGREES OR 4 x PI/6,
WHICH SIMPLIFIES TO 2PI/3 AND SO ON.
SO THIS IS ACTUALLY A VERY HELPFUL DIAGRAM
WHEN YOU'RE FIRST LEARNING HOW TO WORK WITH RADIANS.
THE SECOND KEY ANGLE WOULD BE 45 DEGREES
BEING EQUAL TO PI/4 RADIANS.
SO IF WE HAVE AN ANGLE WITH A MULTIPLE OF 45 DEGREES
IT WILL BE A MULTIPLE OF PI/4 RADIANS.
SO SINCE 90 DEGREES IS 2 x 45 DEGREES, IT WOULD BE 2 x PI/4,
WHICH SIMPLIFIES TO PI/2 AND SO ON.
SO THESE TWO DIAGRAMS MIGHT BE VERY HELPFUL FOR SOME NOTES.
LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE OF FUNCTION VALUES
SINCE WE DO HAVE TIME.
SINCE YOU MAY NOT BE COMFORTABLE WITH RADIANS YET
LET'S GO AHEAD AND CONVERT THIS TO DEGREES.
SO 2PI/3 x 180 DEGREES/PI AND THEN SIMPLIFY,
EQUALS 120 DEGREES.
SO WE KNOW THIS IS EQUAL TO THE TANGENT OF 120 DEGREES.
NOW, AGAIN, ANOTHER WAY TO THINK OF THIS
IS IF WE KNOW THAT PI/6 RADIANS IS EQUAL TO 30 DEGREES
WE COULD TAKE 2PI/3 CONVERT IT TO A DENOMINATOR OF 6
AND THIS WOULD BE 4PI/6, WHICH WOULD EQUAL 4 x 3 DEGREES,
SINCE PI/6 IS 30 DEGREES AND THE RESULT IS THE SAME,
OF COURSE, 120 DEGREES.
OKAY, LET'S GO AHEAD AND FIND THIS FUNCTION VALUE.
WHAT I'M GOING TO DO IS SKETCH IT IN STANDARD POSITION
AND LOOK FOR OUR REFERENCE TRIANGLE.
HERE'S THE INITIAL SIDE OF THE ANGLE.
THE TERMINAL SIDE WOULD BE 90 + 30 DEGREES MORE.
SOMETHING LIKE THIS.
DROP A PERPENDICULAR TO FORM YOUR REFERENCE TRIANGLE.
THIS WAS A 60 DEGREE REFERENCE ANGLE,
SO WE HAVE A 30, 60, 90 RIGHT TRIANGLE.
THEREFORE, WE KNOW THE SIDES WOULD BE SOME MULTIPLE
OF 1, 2, SQUARE ROOT 3,
BUT SINCE WE'RE IN THE SECOND QUADRANT
OUR X COORDINATE WOULD BE NEGATIVE.
SO THE TANGENT OF 120 DEGREES
IS EQUAL TO THE TANGENT OF THIS ANGLE HERE OPPOSITE
OVER ADJACENT EQUALS SQUARE ROOT 3/-1,
WHICH IS EQUAL TO NEGATIVE SQUARE ROOT 3.
SINE OF 3PI/2 RADIANS.
LET'S GO AHEAD AND CONVERT IT TO DEGREES SIMPLIFYING.
WE HAVE 270 DEGREES.
SO WE'LL FIND THE SINE OF 270 DEGREES.
LET'S PLOT THIS ANGLE IN STANDARD POSITION.
THE INITIAL SIDE--TERMINAL SIDE IS HERE.
NOW, THIS IS A QUADRANTAL ANGLE
SO WE CAN'T FORM A REFERENCE TRIANGLE.
WHAT WE CAN DO IS ASSUME THAT THIS IS THE UNIT CIRCLE
AND THIS IS THE POINT OF INTERSECTION
ON THAT UNIT CIRCLE.
THAT WOULD BE THE POINT (0,-1).
REMEMBER THAT SINE DATA IS EQUAL TO Y
IF THE POINT IS ON THE UNIT CIRCLE AND THEREFORE,
THE SINE OF 270 DEGREES IS EQUAL TO -1.
OKAY, I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.