- HERE'S AN APPLICATION PROBLEM
THAT CAN BE SOLVED USING THE LAW OF COSINES.
A PILOT FLIES IN A STRAIGHT PATH FOR 1 HOUR AND 30 MINUTES.
SHE THEN TAKES A COURSE CORRECTION
HEADING 8 DEGREES TO THE LEFT
AND FLIES 2 HOURS IN A NEW DIRECTION.
IF SHE MAINTAINS A CONSTANT SPEED OF 450 MILES PER HOUR,
HOW FAR IS SHE FROM HER STARTING POINT?
SO ACCORDING TO THIS TRIANGLE,
THE PLANE WOULD START OUT IN THIS DIRECTION HERE,
AND THEN TURNS IN THIS DIRECTION HERE,
AND THIS WOULD BE THE ENDING POSITION.
SO THE PLANE STARTS HERE AND STOPS HERE.
AND THE QUESTION IS,
HOW FAR IS SHE FROM HER STARTING POSITION?
SO WE WANT TO DETERMINE THE LENGTH
OF THIS SIDE OF THE TRIANGLE. LET'S GO AHEAD AND CALL IT X.
NOW, ONE THING THAT'S A LITTLE BIT TRICKY ABOUT THIS QUESTION
IS THAT THIS PLANE IS TRAVELING AT 450 MILES PER HOUR
FOR AN HOUR AND A HALF IN ONE DIRECTION
AND 2 HOURS IN ANOTHER.
BUT WE NEED TO DETERMINE THE LENGTH
OF THESE SIDES OF THE TRIANGLE
WHICH WOULD BE THE DISTANCE TRAVELED.
SO FOR EXAMPLE,
IF THE PLANE IS TRAVELING 450 MILES PER HOUR
FOR 1 HOUR, 30 MINUTES OR 1.5 HOURS,
1.5 x 450 WILL GIVE US THE DISTANCE TRAVELED.
SO THIS DISTANCE HERE IS GOING TO BE 675 MILES.
THEN THE PLANE TURNS AND TRAVELS FOR 2 HOURS AT THE SAME SPEED.
WELL, 2 HOURS x 450 MILES PER HOUR
IS GOING TO GIVE US 900 MILES.
AND NOW, THERE'S ONE MORE THING TO CONSIDER.
WHEN SHE'S TRAVELING IN THIS FIRST PATH,
SHE TURNS 8 DEGREES TO THE LEFT AT THIS POINT HERE.
SO IF WE WERE TO EXTEND THIS SIDE HERE,
THIS LITTLE ANGLE HERE IS 8 DEGREES,
THEREFORE THIS LARGE ANGLE IN THE TRIANGLE
IS GOING TO BE 180 DEGREES - 8 DEGREES OR 172 DEGREES.
NOW WITH THE GIVEN INFORMATION OF THIS TRIANGLE,
WE CAN DETERMINE THE LENGTH OF X.
WHEN USING THE LAW OF COSINES,
IF WE'RE GOING TO USE THE MEASURE OF ANGLE "A,"
THAT MEANS THE SIDE OPPOSITE ANGLE "A"
IS GOING TO BE ON THE LEFT SIDE OF THE EQUATION BY ITSELF.
AND IT'S ALSO TRUE
THAT SIDES B AND C WOULD FORM THE GIVEN ANGLE,
IN THIS CASE 172 DEGREES.
SO IF WE WANT TO APPLY THIS FORMULA,
X WILL BE IN THE POSITION OF "A,"
AND B AND C WOULD BE THE OTHER TWO SIDES.
IT DOESN'T MATTER WHICH ONE IS B AND WHICH ONE IS C.
SO WE'RE GOING TO HAVE X SQUARED = 900 SQUARED
+ 675 SQUARED - 2 x 900 x 675 COSINE 172 DEGREES.
NOW, LET'S GO AHEAD AND EVALUATE THIS ON OUR CALCULATOR.
LET'S START BY MAKING SURE THAT WE'RE IN DEGREE MODE,
WHICH WE ARE.
SO IF WE'RE CAREFUL, WE CAN TYPE THIS IN ALL IN ONE STEP.
WE'LL HAVE 900 SQUARED
+ 675 SQUARED - 2 x 900
x 675 x COSINE 172 DEGREES.
AND REMEMBER THIS IS X SQUARED,
SO WE HAVE X SQUARED = 2,468,800.
LET'S JUST CALL IT 0.7.
NOW, TO DETERMINE THE VALUE OF X,
WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION.
AND SINCE X IS A DISTANCE,
WE'RE ONLY WORRIED ABOUT THE PRINCIPAL SQUARE ROOT
OR POSITIVE SQUARE ROOT.
SO WE HAVE THE SQUARE ROOT OF 2,468,800.7,
SO THE PLANE HAS TRAVELED APPROXIMATELY 1,571 MILES.
I HOPE THIS EXAMPLE WAS HELPFUL.