- WE WANT TO CALCULATE THE COMPLEX NUMBER
-2 + 5I RAISED TO THE 4th POWER.
TO DO THIS WE'LL APPLY DE MOIVRE'S THEOREM
GIVEN HERE BELOW,
BUT NOTICE HOW FOR THE FIRST STEP
WE HAVE TO WRITE THE COMPLEX NUMBER, -2 + 5I,
IN POLAR FORM OR THE FORM GIVEN HERE.
SO LET THIS EQUAL Z TO THE 4th.
AND THE FIRST STEP IS TO WRITE Z
WHICH IS EQUAL TO -2 + 5I IN POLAR FORM.
LET'S GO AHEAD AND GRAPH THIS POINT
ON THE COMPLEX COORDINATE PLANE AND THEN DETERMINE R AND THETA.
SO WE'LL MARK OFF -2 ON THE REAL AXIS
AND +5 ON THE IMAGINARY AXIS.
SO THIS WOULD BE THE POINT FOR -2 + 5I
WHERE THE LENGTH OF THIS SEGMENT HERE WOULD BE R,
AND THETA WOULD BE THIS ANGLE HERE
MEASURED FROM THE POSITIVE X AXIS.
LET'S GO AHEAD AND SKETCH A REFERENCE TRIANGLE.
THIS WOULD BE -2 AND THIS WOULD BE +5.
SO TO FIND R WE CAN USE THE EQUATION
R = SQUARE ROOT OF X SQUARED + Y SQUARED.
AND TO FIND THETA WE CAN USE
TANGENT THETA = Y DIVIDED BY X. LET'S FIRST DETERMINE R.
R IS GOING TO BE = THE SQUARE ROOT OF -2 SQUARED + 5 SQUARED.
SO THAT'S JUST GOING TO BE THE SQUARE ROOT OF 4 + 25 OR 29.
SO HERE IS R, BUT NOW WE NEED TO FIND THETA.
WELL, TANGENT THETA WOULD BE EQUAL TO Y DIVIDED BY X OR -5/2
WHICH MEANS THETA IS EQUAL TO ARC TANGENT OF -5/2,
BUT KEEP IN MIND
THAT OUR ANGLE THETA MUST BE IN THE SECOND QUADRANT.
SO NOW WE'LL GO TO THE CALCULATOR,
MAKE SURE WE'RE IN RADIAN MODE, WHICH WE ARE.
AND WE'LL PRESS SECOND TANGENT -5/2.
NOTICE HOW THE CALCULATOR IS GIVING US AN ANGLE
IN THE FOURTH QUADRANT, IT'S APPROXIMATELY -1.1903 RADIANS.
BUT THAT'S OKAY BECAUSE WE CAN USE THIS ANGLE
TO FIND OUR ANGLE IN THE SECOND QUADRANT.
SO THE CALCULATOR GAVE US ANGLE THETA AS APPROXIMATELY -1.1903
WHICH WOULD BE THIS ANGLE HERE IN THE FORTH QUADRANT.
BUT THIS ALSO TELLS US
THAT SINCE TANGENT IS ALSO NEGATIVE IN THE SECOND QUADRANT,
THAT OUR REFERENCE ANGLE MUST BE 1.1903 RADIANS
WHICH MEANS WE CAN FIND OUR ANGLE THETA
BY TAKING PI RADIANS, OR HALF A ROTATION,
AND SUBTRACTING THIS REFERENCE ANGLE.
SO OUR ANGLE IS APPROXIMATELY 1.9513 RADIANS,
WHICH MEANS THE POLAR FORM OF OUR COMPLEX NUMBER WILL BE R,
WHICH IS THE SQUARE ROOT OF 29 x COSINE OF THETA,
WHICH IS APPROXIMATELY 1.9513 + "I" SINE OF 1.9513 RADIANS.
NOW THAT WE HAVE Z WE CAN APPLY DE MOIVRE'S THEOREM
WHERE N = 4 BECAUSE YOU WANT Z TO THE 4th.
LET'S GO AHEAD AND DO THAT ON THE NEXT SLIDE.
SO THIS IS OUR COMPLEX NUMBER Z IN POLAR FORM.
SO IF WE WANT Z TO THE 4th WE'LL RAISE R TO THE 4th POWER.
SO WE'LL HAVE THE SQUARE ROOT OF 29 RAISED TO THE 4th
AND WE'LL ALSO MULTIPLY EACH ANGLE BY 4.
SO WE'LL HAVE COSINE OF 4 x 1.9513 + "I" SINE OF 4 x 1.9513.
WELL, THE SQUARE ROOT OF 29 TO THE 4th POWER
IS THE SAME AS 29 SQUARED, WHICH IS 841,
AND NOW WE'LL HAVE TO USE THE CALCULATOR
TO GET DECIMAL APPROXIMATIONS FOR THESE TRIG FUNCTION VALUES.
SO WE'LL HAVE COSINE 4 x THETA
WHICH IS APPROXIMATELY 0.0488.
LET'S ALSO FIND SINE OF 4 x THETA.
SO THIS PRODUCT HERE WILL BE APPROXIMATELY 0.9988I.
SO WE HAVE 0.0488 + 0.9988I. NOW WE'LL DISTRIBUTE THE 841.
BUT ONE THING TO KEEP IN MIND
IS THAT IF YOU WERE TO MULTIPLY THIS OUT THE LONG WAY,
NOTICE HOW THERE WOULDN'T BE ANY FRACTIONS OR DECIMALS.
SO WHEN WE DISTRIBUTE WE'LL ROUND TO THE NEAREST INTEGER.
SO WE'LL HAVE 841 x 0.0488. SO THE REAL PART WILL BE 41.
AND IF THE IMAGINARY PART, WE'LL HAVE 841 x 0.9988 x "I".
SO WE'LL HAVE + 840I.
SO WE FOUND -2 + 5I RAISED TO THE 4th POWER = 41 + 840I.
NOW, LET'S GO AHEAD AND CHECK THIS ON THE GRAPHING CALCULATOR.
IF THE CALCULATOR IS IN COMPLEX MODE,
RIGHT HERE IT SAYS "A" + BI, THIS IS COMPLEX MODE.
WE CAN CHECK THIS ANSWER BY JUST TYPING IN THE LEFT SIDE.
SO WE HAVE -2 + 5I.
RAISE THIS TO THE 4th POWER
AND NOTICE HOW THIS DOES VERIFY THAT OUR ANSWER IS CORRECT.
41 + 840I IS WHAT WE HAVE AS WELL.
I HOPE YOU FOUND THIS HELPFUL.