- NOW WE'LL TAKE A LOOK AT TWO EXAMPLES OF CONVERTING A POINT
IN RECTANGULAR COORDINATES TO POLAR COORDINATES.
THIS FIRST EXAMPLE IS PRETTY STRAIGHTFORWARD
AND THE SECOND ONE'S A LITTLE MORE CHALLENGING.
THE FIRST STEP WE WANT TO IDENTIFY THE GIVEN COORDINATES
IN RECTANGULAR FORM.
SO THIS POINT WOULD BE THE ORDERED PAIR (4,1),
WHERE X = 4 AND Y = 1.
SO IF WE WANT TO DETERMINE AN ORDERED PAIR
IN POLAR COORDINATES,
WE'D HAVE TO DETERMINE R AND THETA.
AND, REMEMBER, THESE ARE NOT UNIQUE.
THERE'S AN INFINITE NUMBER OF WAYS
TO EXPRESS A POINT IN POLAR COORDINATES.
BUT PROBABLY THE MOST BASIC WAY WOULD BE TO HAVE
A POSITIVE R VALUE AND A POSITIVE THETA.
SO IF WE DID THAT WE COULD EXPRESS THE DISTANCE HERE AS R,
AND THIS HERE WOULD BE THETA.
SO TO DETERMINE THESE VALUES WE COULD SKETCH A TRIANGLE HERE,
LABEL THE OPPOSITE SIDE 1, AND THE ADJACENT SIDE 4.
BUT NORMALLY MOST TEXTBOOKS JUST GIVE YOU
THESE TWO EQUATIONS HERE,
R SQUARED = X SQUARED + Y SQUARED,
AND TANGENT THETA = Y DIVIDED BY X, WHERE X DOESN'T EQUAL ZERO.
BUT REALLY ALL WE'RE DOING
IS USING RIGHT-TRIANGLE TRIGONOMETRY
TO DETERMINE THESE TWO VALUES.
SO LET'S START BY DETERMINING THE VALUE OF R.
WE'RE GOING TO HAVE R SQUARED = X SQUARED + Y SQUARED,
OR 4 SQUARED + 1 SQUARED.
SO WE HAVE R SQUARED = 16 + 1, OR 17.
IF WE SQUARE ROOT BOTH SIDES OF THE EQUATION,
WE HAVE R = + OR - SQUARE ROOT 17.
REMEMBER THAT R CAN BE POSITIVE OR NEGATIVE.
IF R IS POSITIVE, THE TERMINAL SIDE OF THE ANGLE
MUST PASS THROUGH THE POINT,
AND IF R IS NEGATIVE IT WOULD POINT IN THE OPPOSITE DIRECTION.
SO LET'S GO AHEAD AND STICK TO A POSITIVE VALUE OF R,
SO WE'LL HAVE THE SQUARE ROOT OF 17.
THEN TO DETERMINE THETA,
WE'LL USE A TRIG EQUATION INVOLVING TANGENT,
WHERE TANGENT THETA = Y DIVIDED BY X, IN THIS CASE 1/4.
SO NOW IF WE TAKE THE INVERSE TANGENT ON BOTH SIDES,
ON THE LEFT SIDE WE HAVE THETA,
AND ON THE RIGHT SIDE WE HAVE INVERSE TANGENT OF 1/4.
NOW WE HAVE TO DECIDE IF WE WANT TO HAVE THETA
IN TERMS OF DEGREES OR RADIANS.
IT DOESN'T REALLY MATTER.
LET'S GO AHEAD AND CHECK OUR MODE.
RIGHT NOW WE'RE IN DEGREE MODE.
SO LET'S JUST GO AHEAD AND LEAVE IT IN DEGREES.
OF COURSE, WE COULD ALSO CONVERT IT TO RADIANS IF NEEDED.
SO THE SECOND TANGENT, 1 DIVIDED BY 4,
SO THETA IS APPROXIMATELY 14.0 DEGREES.
AND NOTICE HOW 14 DEGREES WOULD BE IN THE FIRST QUADRANT,
PASSING THROUGH THE GIVEN POINT, WHICH IS WHAT WE NEED,
SINCE WE HAVE A POSITIVE VALUE FOR R.
LET'S GO AND TAKE A LOOK AT A SECOND EXAMPLE.
THE GIVEN POINT IS (-2,3).
WE WANT TO WRITE THIS USING POLAR COORDINATES.
AGAIN, THIS MIGHT BE HELPFUL TO VISUALIZE THIS.
IF WE WANT A POSITIVE VALUE FOR R,
THE TERMINAL SIDE OF THETA WOULD HAVE TO PASS THROUGH THIS POINT,
AND THIS SHOULD BE ONE POSSIBLE ANGLE FOR THETA.
SO THIS WOULD BE OUR REFERENCE TRIANGLE THAT WE COULD USE
TO HELP US DETERMINE R AND THETA.
SO WE COULD LABEL THIS -2 AND LABEL THIS 3.
SO THE HYPOTENUSE, OR R SQUARED,
WOULD BE EQUAL TO X SQUARED + Y SQUARED.
SO WE HAVE R SQUARED = 4 + 9, THAT'S 13.
SO WE HAVE R = + OR - THE SQUARE ROOT 13.
NOW LETS SEE IF WE CAN DETERMINE THE COORDINATES IN POLAR FORM
WITH A POSITIVE VALUE FOR R.
SO ACCORDING TO OUR FORMULA, TANGENT THETA = Y DIVIDED BY X,
WHICH IS GOING TO BE 3 DIVIDED BY -2, OR -3/2.
SO WE TAKE THE INVERSE TANGENT ON BOTH SIDES,
WE'RE GOING TO HAVE THETA = INVERSE TANGENT -3/2.
NOW, REMEMBER, THE RANGE FOR INVERSE TANGENT
IS ON THE OPEN INTERVAL FROM -PI/2 TO PI/2.
SO THE CALCULATOR IS NOT GOING TO GIVE US THIS ANGLE HERE
THAT WE WANT.
BUT WE CAN STILL USE IT
TO DETERMINE THE CORRECT ANGLE FOR THETA.
SO LET'S GO TO OUR CALCULATOR
AND WE'LL DETERMINE INVERSE TANGENT -3/2,
AND IT RETURNS -56.3 DEGREES.
LET'S GO AHEAD AND SKETCH THIS TO SEE WHAT'S HAPPENING.
-56.3 DEGREES WOULD HAVE A TERMINAL SIDE HERE
IN THE OPPOSITE DIRECTION.
SO THIS ANGLE DOES HAVE A TANGENT FUNCTION VALUE OF -3/2,
BUT IT'S NOT THE ANGLE THAT WE NEED.
BUT WE COULD USE IT IF WE USED A NEGATIVE VALUE FOR R.
IF WE LET R = -SQUARE ROOT 13,
THEN REMEMBER, THE TERMINAL SIDE OF THETA
WOULD POINT IN THE OPPOSITE DIRECTION OF THE POINT,
WHICH WOULD BE -56.3 DEGREES.
BUT IF WE DID HAVE TO HAVE A POSITIVE VALUE FOR R,
WE CAN STILL USE THIS INFORMATION
TO DETERMINE R THETA.
THIS TELLS US THAT OUR REFERENCE ANGLE IN THE SECOND QUADRANT
WOULD BE 56.3 DEGREES.
SO 180 - 56.3 DEGREES WOULD GIVE US OUR ANGLE THETA,
AND THAT WOULD BE 123.7 DEGREES.
SO EITHER OF THESE WOULD BE CORRECT,
BUT WE DO HAVE TO PAY CLOSE ATTENTION TO OUR ANGLE THETA
BECAUSE OF THE RANGE OF INVERSE TANGENT.
SO I THINK IT'S ALWAYS IMPORTANT TO SKETCH THE POINT
AND VISUALIZE WHAT YOU'RE DOING
RATHER THAN JUST USE THE FORMULAS BLINDLY.
I HOPE YOU HAVE FOUND THIS HELPFUL.