- LETS TAKE A LOOK AT TWO EXAMPLES OF CONVERTING POINTS
IN RECTANGULAR COORDINATES TO POLAR COORDINATES,
AND THIS TIME WE'LL USE RADIANS.
SO THE FIRST STEP IS TO IDENTIFY THE COORDINATES
OF THE GIVEN POINT.
THIS POINT HAS POINTS (3,-3), WHICH TELLS US X = 3 AND Y = -3.
NOW, IF WE WANT TO EXPRESS THIS POINT IN POLAR COORDINATES,
WE WANT TO DETERMINE R AND THETA.
AND, REMEMBER, THERE'S AN INFINITE NUMBER OF WAYS
TO EXPRESS A POINT IN POLAR COORDINATES.
LET'S JUST SAY WE WANT TO HAVE A POSITIVE VALUE FOR R
AND THE LEAST POSITIVE VALUE FOR THETA.
IF R IS POSITIVE THAT MEANS THE TERMINAL SIDE OF THE ANGLE
WOULD HAVE TO PASS THROUGH THE GIVEN POINT.
SO THE TERMINAL SIDE OF THETA WOULD HAVE TO BE HERE.
SO IT'S OFTEN HELPFUL TO SKETCH A REFERENCE TRIANGLE
TO VISUALIZE WHAT'S HAPPENING HERE.
SO THIS WOULD BE OUR REFERENCE TRIANGLE.
WE WOULD LABEL THIS SIDE OF THE REFERENCE TRIANGLE +3,
AND THIS SIDE -3.
NOW, YOU MAY RECOGNIZE THIS REFERENCE TRIANGLE
AS A 45-45-90 RIGHT TRIANGLE, WHICH IT IS.
BUT WE'RE GOING TO ASSUME WE DON'T RECOGNIZE THIS
AND GO AHEAD AND DETERMINE R AND THETA USING THE FORMULAS.
SO, AGAIN, IF WE WANT THE LEAST POSITIVE ANGLE FOR THETA,
IT'S GOING TO BE THE ANGLE THAT HAS THE INITIAL SIDE HERE
AND THE TERMINAL SIDE HERE.
AND THEN R IS GOING TO BE THE LENGTH OF THIS HYPOTENUSE.
SO TO DETERMINE THESE ANGLES WE HAVE THESE FORMULAS
THAT CAN HELP US.
WE HAVE R SQUARED = X SQUARED + Y SQUARED,
WHICH IS JUST THE PYTHAGOREAN THEOREM.
AND THEN WE HAVE THE EQUATION TANGENT THETA = Y DIVIDED BY X,
AS LONG AS X DOESN'T EQUAL ZERO.
SO WE'RE BASICALLY SOLVING THIS PROBLEM
USING RIGHT TRIANGLE TRIGONOMETRY.
LET'S START BY DETERMINING THE VALUE OF R.
WE KNOW THAT R SQUARED = X SQUARED + Y SQUARED.
SO WE'LL HAVE 3 SQUARED + -3 SQUARED.
SO WE'RE GOING TO HAVE R SQUARED = 9 + 9, THAT'S 18.
SO NOW WE CAN SQUARE-ROOT BOTH SIDES OF THE EQUATION.
THIS IS GOING TO GIVE US R EQUALS--WELL, 18 IS 9 x 2.
SO THIS WILL BE + OR (- 3,SQUARE ROOT 2).
AND SINCE WE'RE TRYING TO USE A POSITIVE VALUE FOR R
FOR OUR POLAR COORDINATES,
WE'RE GOING TO USE (+3,SQUARE ROOT 2) FOR R
NOW WE'RE LEFT TO DETERMINE THETA.
NOW LETS GO AHEAD AND DETERMINE THETA.
AGAIN, WE SHOULD RECOGNIZE THETA
BASED UPON THIS ISOSCELES RIGHT TRIANGLE,
BUT WE'LL ASSUME WE DON'T.
WE KNOW THAT TANGENT THETA = Y DIVIDED BY X.
WELL, Y DIVIDED BY X WOULD BE -3 DIVIDED BY 3, OR -1.
SO IF WE TAKE THE INVERSE TANGENT
ON BOTH SIDES OF THE EQUATION,
THETA = INVERSE TANGENT OF -1.
LET'S GO TO THE CALCULATOR AND SEE WHAT ANGLE THIS GIVES US.
THE FIRST STEP IS TO CHECK THE MODE.
LET'S CHANGE THE MODE TO RADIANS, PRESS ENTER,
GO BACK TO THE HOME SCREEN, SECOND TANGENT -1.
NOW, YOU MIGHT NOT RECOGNIZE THIS,
BUT THIS IS ACTUALLY -PI/4 RADIANS.
LET'S GO AHEAD AND WRITE THIS DOWN,
AND THEN WE'LL TAKE A LOOK AT OUR SKETCH AGAIN.
NOW, -PI/4 RADIANS WOULD BE A CORRECT ANGLE FOR THETA
AND NOTICE HOW THE TERMINAL SIDE WOULD PASS THROUGH
THE GIVEN POINT.
BUT IF WE REALLY WANT TO HAVE
THE LEAST POSITIVE ANGLE FOR THETA,
THAT WOULD BE 315 DEGREES,
OR IN RADIANS THAT WOULD BE 7PI/4 RADIANS.
LET'S GO AHEAD AND TAKE A LOOK AT A SECOND EXAMPLE.
THE GIVEN POINT IN RECTANGULAR COORDINATES WOULD BE (-4,-3).
AND, AGAIN, LET'S GO AHEAD AND SEE
IF WE CAN USE A POSITIVE VALUE FOR R,
WHICH MEANS THE TERMINAL SIDE OF THETA
WOULD HAVE TO PASS THROUGH THIS POINT.
SO THIS WOULD BE OUR REFERENCE TRIANGLE,
WHERE WE WOULD LABEL THIS LEG -4, THIS LEG -3,
AND THIS WOULD BE R.
AND, AGAIN, WE'RE TRYING TO DETERMINE R AND THETA
FOR THIS POINT GIVEN IN RECTANGULAR COORDINATES.
SO WE HAVE R SQUARED = -4 SQUARED + -3 SQUARED.
SO WE HAVE R SQUARED = 16 + 9, THAT'S 25.
WELL, THIS ONE COMES OUT NICE.
WE R = + OR - 5.
AGAIN, IF WE WANT TO USE A POSITIVE VALUE FOR R,
THEN WE'RE GOING TO USE +5.
AND THEN IF WE WANT THETA TO BE THE LEAST POSITIVE ANGLE,
THAT WOULD BE THIS ANGLE HERE.
WELL, LET'S USE OUR TRIG EQUATION TANGENT THETA
= Y DIVIDED BY X,
AND SEE WHAT THE CALCULATOR GIVES US.
SO Y DIVIDED BY X WOULD BE -3 DIVIDED BY -4, OR +3/4.
THEREFORE, THETA WOULD BE EQUAL TO INVERSE TANGENT OF 3/4.
NOW, RIGHT AWAY I CAN TELL THE CALCULATOR
IS NOT GOING TO GIVE US THE CORRECT FUNCTION VALUE,
BECAUSE, REMEMBER, THAT THE RANGE OF INVERSE TANGENT
IS FROM ZERO TO PI RADIANS,
OR FROM ZERO TO -PI/2 RADIANS.
BUT IT WILL GIVE US THE INFORMATION WE NEED
TO DETERMINE WHAT THETA WOULD BE.
SO NOW WE'LL PRESS 2nd, TANGENT 3/4,
AND IT'S GOING TO GIVE US AN ANGLE IN THE FIRST QUADRANT,
WHICH IS APPROXIMATELY .64 RADIANS.
WELL, .64 RADIANS WOULD HAVE A TERMINAL SIDE
THAT POINTS IN THE OPPOSITE DIRECTION HERE,
WHICH WOULD BE OKAY IF WE CHANGED THE SINE OF R.
REMEMBER IF R IS NEGATIVE,
THE TERMINAL SIDE OF THETA WOULD POINT IN THE OPPOSITE DIRECTION.
SO IF WE USE (-5,.64 RADIANS),
THIS WOULD BE THE SAME POINT IN POLAR COORDINATES.
BUT IF WE REALLY WANT TO USE A POSITIVE VALUE FOR R,
THIS .64 RADIANS WOULD BE THE REFERENCE ANGLE
IN THE THIRD QUADRANT.
SO OUR ANGLE THETA IN RED HERE
WOULD BE EQUAL TO PI + .64 RADIANS.
SO 3.14 + .64 WOULD GIVE US APPROXIMATELY 3.78 RADIANS.
AGAIN, BOTH OF THESE REPRESENT THE SAME POINT.
ONE HAS A POSITIVE VALUE OF R,
AND ONE HAS A NEGATIVE VALUE FOR R.
I HOPE YOU HAVE FOUND THIS HELPFUL.