- WE WANT TO WRITE THE COMPLEX NUMBERS IN CARTESIAN FORM
IN EXPONENTIAL FORM WITH THE FORM
R x E RAISED TO THE POWER OF "I" THETA.
WE ALSO WANT THETA TO BE IN THIS INTERVAL HERE.
SO WE WANT TO FIND R, AND WE WANT TO FIND THETA.
NOW, WE CAN USE THESE FORMULAS HERE IF NEEDED,
BUT MANY TIMES,
IF WE PLOT THE POINT ON THE COMPLEX COORDINATE PLANE,
WE CAN FIND SOME OF THE INFORMATION THAT WE NEED.
SO LET'S GO AHEAD AND PLOT THE NUMBER 2I
ON THE COMPLEX COORDINATE PLANE.
BECAUSE 2I IS AN IMAGINARY NUMBER,
IT'S GOING TO BE ON THE IMAGINARY AXIS UP TWO UNITS.
SO THE LENGTH OF THIS SEGMENT HERE WOULD BE THE RADIUS,
WHICH WOULD BE TWO, AND THEN ANGLE THETA,
MEASURED FROM THE POSITIVE X AXIS WOULD BE PI/2 RADIANS,
WHICH MEANS 2I WRITTEN IN EXPONENTIAL FORM
WOULD BE 2 x E RAISED TO THE POWER OF "I" x PI/2 OR PI/2I.
NEXT, WE HAVE 3 - 3I.
LET'S GO AHEAD AND PLOT THIS COMPLEX NUMBER.
SO +3 ON THE REAL AXIS, -3 ON THE IMAGINARY AXIS.
SO THIS POINT HERE WOULD BE OUR COMPLEX NUMBER.
THE LENGTH OF THIS SEGMENT HERE WOULD BE R.
LET'S GO AHEAD AND SKETCH THE REFERENCE TRIANGLE FOR THIS ONE.
WE COULD LABEL THIS LEG +3, LABEL THIS LEG -3,
BECAUSE BOTH LEGS IN THIS RIGHT TRIANGLE
HAVE A LENGTH OF THREE UNITS.
WE SHOULD RECOGNIZE THIS AS A 45/45/90 REFERENCE TRIANGLE
WHERE THE REFERENCE ANGLE WOULD BE PI/4 RADIANS.
THIS IS HELPFUL IN TWO WAYS.
FIRST, WE SHOULD RECOGNIZE THAT ANGLE THETA, THIS ANGLE HERE,
WOULD BE 2PI RADIANS - PI/4 RADIANS OR 7PI/4 RADIANS.
SO THAT'S THETA.
NORMALLY, WHEN WE HAVE A 45/45/90 REFERENCE TRIANGLE,
WE USE THE LENGTHS OF 1, 1 SQUARE ROOT 2,
BUT NOTICE HOW IF WE MULTIPLY EACH SIDE BY 3,
WE WOULD HAVE OUR REFERENCE TRIANGLE,
WHICH MEANS THE HYPOTENUSE OF THIS REFERENCE TRIANGLE
OR R = 3 SQUARE ROOT 2.
AGAIN, WE COULD USE THESE FORMULAS HERE,
BUT IF YOU RECOGNIZE THIS, IT'LL SAVE US SOME TIME.
SO 3 - 3I = 3 SQUARE ROOT 2 x E RAISED TO THE POWER OF "I"
x THETA OR 7PI/4 x "I".
NOW, LET'S TAKE A LOOK AT OUR LAST EXAMPLE.
WE HAVE -4 + 3I.
SO WE'LL MARK OFF -4 ON THE REAL AXIS
AND +3 ON THE IMAGINARY AXIS. SO HERE'S OUR COMPLEX NUMBER.
THIS SEGMENT HERE WOULD BE R.
AGAIN, LET'S GO AHEAD AND SKETCH THE REFERENCE TRIANGLE.
THIS WOULD BE -4. THIS WOULD BE +3.
NOW, IN THIS EXAMPLE,
WE SHOULD RECOGNIZE THAT THIS RIGHT TRIANGLE
IS A 3/4/5 RIGHT TRIANGLE, AND THEREFORE, R EQUALS 5.
IF WE DIDN'T RECOGNIZE THIS,
WE COULD USE THE PYTHAGOREAN THEOREM
OR THIS FORMULA HERE TO DETERMINE R.
WE STILL HAVE TO FIND THETA, WHICH WOULD BE THIS ANGLE HERE.
WELL, WE KNOW TANGENT THETA MUST EQUAL Y DIVIDED BY X.
SO IN THIS CASE, WE'D HAVE TANGENT THETA = -3/4.
THEREFORE, THETA = ARCH TANGENT OF -3/4.
WE'RE NOT GOING TO FIND THIS TRIG FUNCTION VALUE
ON THE UNIT CIRCLE OR ON OUR REFERENCE TRIANGLES.
SO WE'LL HAVE TO USE THE CALCULATOR
TO GET A DECIMAL APPROXIMATION, BUT BEFORE WE DO THIS,
RECOGNIZE THAT OUR ANGLE THETA MUST BE IN THE SECOND QUADRANT.
LET'S MAKE SURE THAT WE ARE IN RADIAN MODE. WE ARE.
SO WE'LL PRESS 2nd TANGENT -3/4.
NOTICE HOW THE CALCULATOR
IS GIVING US THE ANGLE IN THE FOURTH QUADRANT,
BUT THAT'S OKAY,
BECAUSE WE CAN USE THIS TO FIND OUR ANGLE THETA.
THE CALCULATOR IS GIVING US APPROXIMATELY -0.6435 FOR THETA,
AND THAT WOULD BE THIS ANGLE HERE,
WHICH MEANS OUR REFERENCE ANGLE MUST BE 0.6435 RADIANS HERE.
SO WE CAN FIND THETA BY TAKING PI RADIANS OR HALF A ROTATION
AND SUBTRACTING OUR REFERENCE ANGLE.
SO THETA IS APPROXIMATELY PI - 0.6435 RADIANS.
SO OUR THETA IS APPROXIMATELY 2.4981 RADIANS.
NOW THAT WE HAVE R AND THETA,
WE CAN WRITE THE COMPLEX NUMBER IN EXPONENTIAL FORM.
IT'S GOING TO BE APPROXIMATELY
5 x E RAISED TO THE POWER OF 2.4981 x "I".
OKAY. HOPE YOU FOUND THESE THREE EXAMPLES HELPFUL.