- WELCOME TO THE SECOND VIDEO ON THE APPLICATION
OF RIGHT TRIANGLE TRIGONOMETRY.
THE GOAL OF THIS VIDEO IS TO SOLVE THE APPLICATION PROBLEMS
USING RIGHT TRIANGLE TRIGONOMETRY.
RIGHT TRIANGLES CAN BE USED TO FIND THE DISTANCES
IF THE ANGLE OF ELEVATION OR DEPRESSION IS KNOWN.
SO THIS HORIZONTAL LINE REPRESENTS THE LINE OF SIGHT.
LOOKING UPWARD THIS WOULD BE THE ANGLE OF ELEVATION
OR LOOKING DOWNWARD THIS WOULD BE THE ANGLE OF DEPRESSION.
SIMILARLY BEARINGS WHICH ARE USED IN NAVIGATION
CAN BE USED TO DETERMINE DISTANCES.
BEARINGS ARE MEASURED FROM DUE NORTH
OR IN TERMS OF NORTH, EAST, SOUTH OR WEST.
SO HERE'S AN EXAMPLE.
HERE'S DUE NORTH.
NORTH, 50 DEGREES EAST WOULD BE REPRESENTED
BY THIS ANGLE HERE.
NOTICE HOW IT'S MEASURED CLOCKWISE FROM DUE NORTH
AND THAT'S IMPORTANT BECAUSE DIRECTION IS NOT MENTIONED
IT IS ASSUMED THE BEARINGS ARE GIVEN AS ANGLES
ROTATED CLOCKWISE FROM THE NORTH.
AS ANOTHER EXAMPLE LET'S SAY THIS WAS 30 DEGREES.
I COULD SAY A BEARINGS OF 210 DEGREES,
180 + 30 OR I COULD SAY SOUTH, 30 DEGREES WEST.
TWO WAYS TO NOTE THE SAME DIRECTION.
LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE OF PROBLEMS.
ONE'S EYESIGHT IS 5 FEET 6 INCHES FROM THE GROUND.
WHEN HE STANDS 100 FEET AWAY FROM THE FLAG POLE
THE ANGLE OF ELEVATION IS 45 DEGREES, 48 MINUTES.
WHAT IS THE HEIGHT OF THE FLAG POLE?
SO LET'S GO AHEAD AND SKETCH WHAT'S HAPPENING HERE.
IF THIS IS JUAN THAN LET'S CALL THIS THE LINE OF SIGHT.
THIS WOULD BE 5 FEET, 6 INCHES.
LET'S CONVERT THIS TO A DECIMAL.
SO THIS WOULD BE 5.5 FEET.
AND HE'S STANDING 100 FEET AWAY FROM THE FLAG POLE.
SO OVER HERE 100 FEET WE HAVE A FLAG.
AND THE ANGLE OF ELEVATION,
THIS ANGLE HERE IS 45 DEGREES, 48 MINUTES.
LET'S GO AHEAD AND CONVERT THIS 48 MINUTES TO A DECIMAL.
SO WE WOULD TAKE 48 AND DIVIDE BY 60
AND THAT'S GOING TO GIVE US .8.
SO WE HAVE 45.8 DEGREES FOR OUR ANGLE OF ELEVATION.
AND THE QUESTION IS,
WHAT IS THE HEIGHT OF THE FLAG POLE?
SO WE CAN FIGURE OUT THIS DISTANCE HERE.
LET'S JUST CALL IT (Y).
AND THEN WE CAN'T FORGET
THAT WE DO HAVE TO ADD THIS EXTRA 5.5 FEET
TO FIND THE TOTAL HEIGHT OF THE FLAG POLE.
SO WE HAVE A RIGHT TRIANGLE HERE.
WHAT WE CAN SEE THAT THIS IS THE OPPOSITE SIDE
AND THE ADJACENT SIDE TO THE GIVEN ANGLE.
SO WE CAN SAY THE TANGENT OF 45.8 DEGREES MUST EQUAL Y/100.
SO IF WE PUT THIS OVER 1 WE CAN JUST PERFORM CROSS PRODUCTS.
1 x (Y) WOULD EQUAL (Y).
AND (Y) WILL EQUAL TANGENT 45.8 DEGREES x 100.
MAKE SURE YOUR CALCULATOR IS IN DEGREE MODE.
TIMES 100 AND 102.83 FEET.
NOW BE CAREFUL.
THAT'S NOT THE HEIGHT OF THE FLAG
BECAUSE WE STILL HAVE TO ADD 5.5 FEET
FROM HIS LINE OF SIGHT TO THE GROUND.
SO THE TOTAL HEIGHT OF THE FLAG POLE WOULD BE 108.33 FEET.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE PROBLEM.
A SHIP TRAVELS 55 KILOMETERS ON A BEARING OF 27 DEGREES,
THEN TRAVELS ON A BEARING OF 117 DEGREES FOR 120 KILOMETERS.
FIND THE DISTANCE TRAVELED FROM THE STARTING POINT
TO THE ENDING POINT.
SO THE FIRST THING I NOTICE THERE'S NO DIRECTION GIVEN
WITH THESE ANGLES.
SO WE KNOW THAT THEY'RE CLOCKWISE FROM DUE NORTH.
SO IF WE SEE OUR STARTING POINT IS HERE
AND DUE NORTH IS HERE WE WOULD ROTATE CLOCKWISE 27 DEGREES
FOR A DISTANCE OF 55 KILOMETERS.
SO THIS WILL BE 55 KILOMETERS AND THIS WOULD BE 27 DEGREES.
NOW, FROM THIS POINT THIS IS DUE NORTH.
NOW WE'LL ROTATE 117 DEGREES FOR 120 KILOMETERS.
SO 117 DEGREES WOULD BE ROUGHLY 90 AND THEN ABOUT 27 MORE
AND FOR A DISTANCE OF 120 KILOMETERS.
LET'S CALL THIS 120 KILOMETERS AND THIS WOULD BE 117 DEGREES.
AND THE QUESTION IS WHAT IS THE DISTANCE FROM THIS POINT
TO THE STARTING POINT SO WE'LL CALL THIS (X).
SO WE NEED TO FIGURE OUT SO MORE ANGLES FOR THIS TRIANGLE.
AND THE FIRST THING WE SHOULD NOTICE
IS THAT THESE TWO RED LINES WOULD BE PARALLEL
BECAUSE THEY'RE BOTH DUE NORTH.
SO IF THEY'RE BOTH DUE NORTH THESE TWO INTERIOR ANGLES
WOULD BE SUPPLEMENTARY OR WOULD ADD TO 180 DEGREES.
SO 180 - 27 COULD GIVE US 153 DEGREES FOR THIS ANGLE.
AND WE KIND OF LUCKED OUT HERE BECAUSE IF YOU ADD 153 + 117
THAT'S GOING TO EQUAL 270 DEGREES.
WELL IF THESE TWO ADD TO 270
AND THESE THREE ANGLES WOULD EQUAL 360
THIS MAKES THIS ANGLE RIGHT HERE A RIGHT ANGLE.
SO WE DO HAVE A RIGHT TRIANGLE.
AND SINCE WE HAVE A RIGHT TRIANGLE
WE KNOW THAT THREE SIDES OF A RIGHT TRIANGLE CAN RELATE
TO ONE ANOTHER FROM THE PYTHAGOREAN THEOREM.
SO PYTHAGOREAN THEOREM STATES THAT "C" SQUARED
IS EQUAL TO "A" SQUARED + "B" SQUARE
WHERE "C" SQUARED IS THE HYPOTENUSE.
SO WE HAVE X SQUARED MUST EQUAL 55 SQUARED + 120 SQUARED.
SO X SQUARED WILL EQUAL, GO TO OUR CALCULATOR,
55 SQUARED + 120 SQUARED, 17,425.
AND THEN WE'LL TAKE THE PRINCIPAL SQUARE ROOT.
AND WE CAN SEE THAT IT'S APPROXIMATELY 132 KILOMETERS.
AND THAT WILL DO IT FOR THIS VIDEO.
THANK YOU FOR WATCHING AND HAVE A NICE DAY.