Title: Geometry Applications: Right Triangles

Title: Geometry Applications: Right Triangles

[Music]

[Music]

IN THE ANCIENT PORT CITY OF CORINTH

SHIPPING WAS AN IMPORTANT PART OF LIFE IN GREECE.

BECAUSE OF ITS MOUNTAINOUS TERRAIN, TRANSPORTING GOODS

AROUND THE COUNTRY BY SHIP WAS NECESSARY

AND SHIPPING EVOLVED OVER TIME.

AN EARLY POPULAR KIND OF SHIP HAD A RECTANGULAR SAIL

AND ALSO RELIED ON OARSMEN TO MOVE THE SHIP FORWARD.

WHEN THE WIND WAS BLOWING IN THE RIGHT DIRECTION,

THE SAIL PROVIDED A QUICK WAY OF

PROPELLING THE SHIP FORWARD.

BUT THIS ONLY WORKED WHEN THE WIND

CAME FROM A PARTICULAR DIRECTION.

COMING FROM A DIFFERENT DIRECTION

PROVIDED NO MOVEMENT FORWARD, AND IN SUCH A CASE

THE OARSMEN PROVIDED THE SHIP'S MOTION.

THESE TYPES OF SHIPS WERE USED MOSTLY FOR MILITARY

PURPOSES SINCE THE OARSMEN WERE ALSO SOLDIERS.

BUT SOMETHING DIFFERENT WAS NEEDED

FOR TRANSPORTING GOODS.

CORINTH WAS A THRIVING MERCHANT CITY

WITH A CONSTANT FLOW OF MERCHANT SHIPS,

AS WERE OTHER GREEK PORT CITIES LIKE ATHENS.

SUCH SHIPS NEEDED ROOM FOR THE GOODS BEING TRANSPORTED.

SO THE EXTRA ROOM TAKEN UP BY OARSMEN

WAS A DISADVANTAGE.

SO OVER TIME THE RECTANGULAR SHAPED SAIL

GAVE RISE TO THE TRIANGULAR SHAPED SAIL

AND WHAT WE NOWADAYS CALL A SAILBOAT.

A SAILBOAT HAS MANY ADVANTAGES

TO THE SQUARE SAIL SHIP.

NOT ONLY DOES THE SAILBOAT MOVE FORWARD

WHEN THE WIND IS BLOWING IN THE RIGHT DIRECTION,

BUT THE SAILBOAT OFFERS THE ABILITY

TO MOVE FORWARD EVEN WHEN THE WIND

IS BLOWING FROM THE SIDE OR EVEN FROM THE FRONT.

OVER TIME, AS SAILORS SOUGHT TO TRAVEL GREATER DISTANCES,

LARGER SHIPS THAT COMBINED RECTANGULAR

AND TRIANGULAR SAILS WERE THE LOGICAL NEXT STEP.

IT WAS THESE SHIPS THAT CONQUERED THE SEAS.

BUT IT WAS THE USE OF THE TRIANGULAR SAIL THAT MADE

ALL OF THIS INNOVATION IN SHIP BUILDING POSSIBLE.

TRIANGULAR SAILS MADE SHIPS MORE VERSATILE.

LET'S TAKE A CLOSER LOOK.

THE PRINCIPLE THAT ALLOWS A SAILBOAT

TO MOVE FORWARD EVEN AGAINST THE WIND

IS THE SAME ONE THAT ALLOWS AN AIRPLANE TO FLY.

AIRPLANE WINGS ARE USUALLY TRIANGULAR SHAPED

AND THEY HAVE A CURVED EDGE ALONG THE SIDE

THAT SLICES THROUGH THE WIND.

AS THE PLANE'S WING MOVES AGAINST THE WIND,

THE AIR PRESSURE ABOVE THE WING IS LESS THAN

THE AIR PRESSURE BELOW THE WING.

AND THE DIFFERENCE IN PRESSURE PUSHES THE WING

AND THE AIRPLANE UP.

WITH THE SAILBOAT THE SAME PRINCIPLE APPLIES

BUT IN A DIFFERENT MANNER.

SUPPOSE THE WIND IS COMING FROM THE SIDE.

THE SAILOR WILL ORIENT THE BOAT

AT AN ANGLE TO THE DIRECTION OF THE WIND.

THE WIND GOES ACROSS THE SAIL BUT THE CURVED SIDE

IS THE ONE THAT GETS LESS AIR PRESSURE

SO THE SAIL AND THE SHIP IS PUSHED IN THIS DIRECTION.

THE FORCE ON THE BOAT IS AT AN ANGLE

RELATIVE TO THE SAILBOAT.

BUT WE CAN CONSTRUCT A RIGHT TRIANGLE

TO SEE THE PARALLEL AND PERPENDICULAR

COMPONENTS OF THE FORCE.

NOTICE THAT THE LEG OF THE RIGHT TRIANGLE

PARALLEL TO THE SAILBOAT

WILL PUSH THE SAILBOAT FORWARD.

THE OTHER COMPONENT OF THE FORCE

PUSHES THE SHIP SIDEWAYS.

THE SHIP CAN COMPENSATE FOR THE SIDEWAYS MOTION

BY POINTING THE SAIL IN THE OPPOSITE DIRECTION.

THIS CREATES A SIDEWAYS FORCE IN THE OPPOSITE

DIRECTION AS WELL AS A PARALLEL FORCE THAT

CONTINUES PUSHING THE SAILBOAT FORWARD.

THIS ZIGZAGGING MOTION THROUGH THE WIND,

A SAILING TECHNIQUE CALLED TACKING,

ALLOWS THE SAILBOAT TO MOVE FORWARD

UNDER ANY WIND CONDITIONS.

THE SHAPE OF THE SAIL IS TRIANGULAR.

BUT THE IDEAL TYPE OF TRIANGLE

TO USE FOR THE SAIL IS A RIGHT TRIANGLE.

LET'S USE THE TI-NSPIRE TO EXPLORE THE

PROPERTIES OF RIGHT TRIANGLES

THAT MAKE IT IDEAL FOR THE SHAPE OF A SAIL.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPHS AND GEOMETRY WINDOW.

YOU'LL BE CONSTRUCTING A TRIANGLE

TO SIMULATE THE SHAPE OF A SAIL.

PRESS MENU AND UNDER "POINTS AND LINES" SELECT SEGMENT.

MOVE THE NAV PAD TOWARD THE LOWER

MIDDLE PART OF THE SCREEN.

CREATE A LONG SEGMENT THAT COVERS MOST OF THE

HORIZONTAL DISTANCE OF THE SCREEN.

PRESS ENTER TO DEFINE THE FIRST ENDPOINT

OF THE SEGMENT.

PRESS THE RIGHT ARROW KEY AND YOU WILL SEE

THE SEGMENT TAKE SHAPE.

PRESS ENTER TO DEFINE THE SECOND ENDPOINT.

THIS SEGMENT WILL INCLUDE THE BASE OF THE TRIANGLE

SO USE THE LEFT ARROW TO MOVE THE POINTER

ABOVE THE SEGMENT YOU JUST CREATED,

STOPPING AT A POINT BEFORE YOU REACH THE ENDPOINT.

PRESS ENTER.

THIS ADDS A POINT TO THE SEGMENT.

THIS WILL BE ONE OF THE VERTICES OF THE TRIANGLE.

USE THE UP AND RIGHT ARROWS TO MOVE THE POINTER

TO CREATE ONE OF THE SIDES OF THE TRIANGLE.

PRESS ENTER TO DEFINE THE ENDPOINT OF THE SEGMENT.

IMMEDIATELY PRESS ENTER AGAIN TO CONSTRUCT THE

ENDPOINT OF THE SECOND SIDE OF THE TRIANGLE.

USE THE DOWN AND RIGHT ARROWS TO MOVE THE POINTER

TO CREATE THE SECOND SIDE OF THE TRIANGLE.

STOP WHEN YOU REACH THE HORIZONTAL LINE.

MAKE SURE THE POINTER IS TO THE LEFT OF THE

ENDPOINT OF THE LONG HORIZONTAL SEGMENT

BUT ALSO HOVERING OVER THE SEGMENT.

PRESS ENTER.

YOU NOW HAVE A TRIANGLE

RESTING ON THE LONG HORIZONTAL LINE.

YOUR SCREEN SHOULD LOOK LIKE THIS.

ASSUME THAT THIS ILLUSTRATION REPRESENTS

A SAIL ATTACHED TO THE MAST.

WE WILL MEASURE TWO SIDES OF THE TRIANGLE -

THE BASE AND ONE OF THE SLANTED SIDES.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

USE THE NAV PAD TO SELECT THE LENGTH OF THE BASE.

YOU DON'T WANT TO MEASURE THE LENGTH OF THE WHOLE

SEGMENT, JUST THE PORTION BETWEEN THESE TWO POINTS.

SO USE THE NAV PAD TO MOVE THE POINTER TO ONE ENDPOINT

AND PRESS ENTER.

THEN MOVE THE POINTER TO THE OTHER ENDPOINT

AND PRESS ENTER AGAIN.

YOU WILL SEE THE LENGTH MEASUREMENT.

MOVE THE POINTER TO BELOW THE BASE

AND PRESS ENTER ONCE MORE TO PLACE THE MEASUREMENT.

NOW USE THE NAV PAD TO MOVE THE POINTER

TO THE SECOND SIDE.

SINCE THERE IS NO AMBIGUITY ABOUT

WHICH LENGTH YOU'RE MEASURING,

SIMPLY HOVER THE POINTER OVER THE SEGMENT.

YOU WILL SEE THE MEASUREMENT APPEAR.

PRESS ENTER ONCE TO SELECT IT.

MOVE THE POINTER TO THE LEFT AND THEN

PRESS ENTER ONCE MORE TO PLACE THE MEASUREMENT.

WE WANT TO SET THE MEASUREMENTS OF THE BASE

AND TRIANGULAR SIDE TO A FIXED LENGTH.

PRESS ESCAPE TO EXIT MEASUREMENT MODE.

MOVE THE POINTER ABOVE THE MEASUREMENT

OF THE SECOND SIDE.

PRESS ENTER TWICE TO EDIT THE TEXT OF THE MEASUREMENT.

CHANGE THE LENGTH TO SIX AND PRESS ENTER.

REPEAT FOR THE LENGTH OF THE BASE.

CHANGE ITS VALUE TO FOUR AND PRESS ENTER AGAIN.

WE WANT TO MAKE SURE THAT THESE LENGTHS DON'T CHANGE

EVEN IF THE SHAPE OF THE TRIANGLE CHANGES.

MOVE THE POINTER ABOVE ONE OF THE MEASUREMENTS.

PRESS CONTROL AND THE MENU KEY.

AT THE DROP DOWN MENU SELECT ATTRIBUTES.

USE THE DOWN ARROW TO SELECT THE ICON

THAT LOOKS LIKE AN OPEN LOCK.

PRESS THE RIGHT ARROW TO CHANGE THE ICON

TO A CLOSED LOCK.

PRESS ENTER.

THIS LOCKS IN THE LENGTH OF THE TRIANGLE SIDE.

REPEAT THIS FOR THE BASE OF THE TRIANGLE.

NOW AS YOU MANIPULATE THE TRIANGLE

THE TWO SIDES WITH FIXED LENGTHS DO NOT CHANGE.

WE WANT TO MEASURE THE AREA OF THIS TRIANGULAR SAIL.

THE FORMULA FOR THE AREA OF A TRIANGLE IS:

AREA = 1/2 (BASE) TIMES HEIGHT.

WE KNOW THE LENGTH OF THE BASE

BUT WE DON'T KNOW THE LENGTH OF THE HEIGHT.

IN FACT, THE HEIGHT IS DEFINED AS

THE LINE PERPENDICULAR TO THE BASE

THAT INTERSECTS THE VERTEX OPPOSITE THE BASE.

LET'S CONSTRUCT THIS LINE SEGMENT.

PRESS MENU AND UNDER CONSTRUCTION

SELECT PERPENDICULAR.

MOVE THE POINTER ABOVE THE VERTEX

THAT FACES THE BASE OF THE TRIANGLE.

PRESS ENTER.

NOW USE THE DOWN ARROW TO MOVE THE POINTER

UNTIL IT INTERSECTS THE BASE OF THE TRIANGLE.

PRESS ENTER ONCE TO DEFINE THE

PERPENDICULAR SEGMENT AND PRESS ENTER AGAIN

TO ADD AN INTERSECTION POINT TO THE BASE.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

NOW MEASURE THE LENGTH OF THE HEIGHT.

PRESS MENU AND UNDER MEASUREMENT SELECT LENGTH.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE SEGMENT THAT DEFINES THE HEIGHT.

YOU SHOULD SEE A MEASUREMENT

FOR THE LENGTH OF THE SEGMENT.

PRESS ENTER ONCE TO SELECT THE MEASUREMENT.

MOVE THE POINTER TO THE SIDE OF THE SEGMENT AND

PRESS ENTER AGAIN TO PLACE THE MEASUREMENT ON-SCREEN.

YOU NOW HAVE ALL THE MEASUREMENTS YOU NEED

TO CALCULATE THE AREA OF THE TRIANGLE.

SINCE YOU WILL BE MANIPULATING THE TRIANGLE

AND CHANGING THE AREA, CREATE A FORMULA

FOR TRACKING THE CHANGING VALUES OF THE AREA.

PRESS MENU AND UNDER ACTIONS SELECT TEXT.

MOVE THE POINTER BELOW THE TRIANGLE

TO A CLEAR PART OF THE SCREEN.

PRESS ENTER ONCE TO SEE A TEXT CURSOR.

INPUT THE EXPRESSION ONE HALF B TIMES H.

PRESS ENTER.

YOU WANT TO LINK THE FORMULA FOR AREA

TO THE SPECIFIC VALUES FROM THE TRIANGLE.

PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.

MOVE THE POINTER SO THAT IT HOVERS

OVER THE FORMULA YOU JUST INPUT.

PRESS ENTER.

NOTICE THE PROMPT THAT ASKS FOR THE VALUE OF B.

MOVE THE POINTER SO THAT IT HOVERS OVER

THE MEASUREMENT OF THE BASE.

PRESS ENTER.

NOW NOTICE THAT THE PROMPT ASKS FOR THE VALUE OF H.

MOVE THE POINTER ABOVE THE VALUE FOR THE HEIGHT.

PRESS ENTER.

YOU'LL SEE THE VALUE OF THE AREA.

MOVE THE POINTER SO THAT IT IS NEAR THE AREA FORMULA.

PRESS ENTER TO PLACE THE VALUE NEXT TO IT.

FINALLY, YOU WANT TO TRACK THE ANGLE

FORMED BY THE TWO SIDES OF THE TRIANGLE

WHOSE MEASUREMENTS YOU MEASURED.

PRESS MENU AND UNDER MEASUREMENT SELECT ANGLE.

MOVE THE POINTER ABOVE THE UPPER VERTEX

OF THE TRIANGLE.

PRESS ENTER.

USE THE DOWN ARROW TO MOVE THE POINTER

TO THE CORNER VERTEX.

PRESS ENTER AGAIN.

USE THE RIGHT ARROW TO MOVE THE POINTER

TO THE LAST VERTEX.

PRESS ENTER ONE MORE TIME.

YOU SHOULD NOW SEE THE ANGLE MEASURE.

PRESS ESCAPE TO EXIT MEASUREMENT MODE.

SELECT AND MOVE THE UPPER VERTEX OF THE TRIANGLE.

NOTE HOW THE ANGLE MEASURE AND THE AREA CHANGE.

SINCE WE FIXED THE VALUES FOR THE BASE

AND ONE OF THE SIDES OF THE TRIANGLE,

THEN THE REASON THAT THE AREA CHANGES

IS BECAUSE THE HEIGHT OF THE TRIANGLE CHANGES.

CREATE A TWO COLUMN DATA TABLE.

THE XY DATA THAT YOU GENERATE WILL BE USED

TO CREATE A SCATTERPLOT.

THE X DATA WILL BE THE ANGLE MEASURE

AND THE Y DATA WILL BE THE AREA OF THE TRIANGLE.

ON A SHEET OF PAPER LABEL ONE COLUMN "ANGLE"

AND THE OTHER COLUMN "AREA".

GENERATE 20 DATA POINTS BY MANIPULATING THE TRIANGLE

FOR DIFFERENT ANGLES.

INCLUDE A VARIETY OF ANGLES.

PAUSE THE VIDEO TO GENERATE YOUR DATA.

YOU WILL NOW TRANSFER THIS DATA SET TO A SPREADSHEET.

PRESS THE HOME KEY AND CREATE A LIST

AND SPREADSHEET WINDOW.

USE THE UP ARROW TO MOVE THE CURSER

TO THE CELL AT THE TOP OF COLUMN A.

INPUT THE LABEL "ANGLES".

PRESS THE TAB KEY TO GO TO COLUMN B.

INPUT THE LABEL "AREA".

USE THE NAV PAD TO MOVE THE CURSER TO CELL A1.

INPUT THE XY DATA YOU GENERATED.

PAUSE THE VIDEO TO INPUT THE DATA.

CREATE A SCATTERPLOT.

PRESS THE HOME KEY AND CREATE A

GRAPHS AND GEOMETRY WINDOW.

BY DEFAULT THE GRAPH WINDOW IS FOR A FUNCTION GRAPH.

PRESS MENU AND UNDER "GRAPH TYPE" SELECT SCATTERPLOT.

INPUT THE COLUMN HEADINGS YOU USED

FOR COLUMNS A AND B.

USE THE COLUMN HEADING FOR A IN THE FLD MARKED X.

PRESS THE DOWN ARROW

AND INPUT THE COLUMN HEADING FOR B.

PRESS ENTER.

YOU'LL SEE SOME OF THE DATA GRAPHED.

TO SEE ALL OF IT, PRESS MENU

AND UNDER WINDOW SELECT ZOOM DATA.

YOU'LL SEE A GRAPH OF THE DATA THAT YOU CREATED

AND IT SHOULD LOOK LIKE THIS.

THE SHAPE OF THE GRAPH IS A DOWNWARD FACING PARABOLA

WITH ONE POINT REPRESENTING THE MAXIMUM AREA.

TRY TO IDENTIFY THE ANGLE MEASURE ALONG THE X AXIS

THAT CORRESPONDS TO THE MAXIMUM.

YOU'LL SEE THAT THE MAXIMUM AREA OF A TRIANGLE

OCCURS WHEN THE ANGLE BETWEEN THE

BASE AND ONE OF THE SIDES IS 90 DEGREES.

IN OTHER WORDS, THE RIGHT TRIANGLE IS THE TRIANGLE

WITH THE MAXIMUM AREA FOR A GIVEN BASE AND SIDE LENGTH.

KNOWING THAT A TRIANGULAR SHAPE IS BEST SUITED

FOR SAILING AGAINST THE WIND, AND WANTING THE SAIL

TO HAVE THE MAXIMUM AREA FOR THE WIND TO BLOW ON,

THEN IT MAKES SENSE FOR A SAILBOAT TO HAVE A SAIL

IN THE SHAPE OF A RIGHT TRIANGLE.

FOR ANY RIGHT TRIANGLE, THE SIDES THAT DEFINE THE

RIGHT ANGLE ARE CALLED THE LEGS OF THE RIGHT TRIANGLE.

THE SIDE OPPOSITE THE RIGHT ANGLE

IS CALLED THE HYPOTENUSE.

FOR ANY RIGHT TRIANGLE, IF WE LABEL THE LEGS A AND B

AND THE HYPOTENUSE C,

THEN ACCORDING TO THE PYTHAGOREAN THEOREM,

A SQUARED PLUS B SQUARED EQUALS C SQUARED.

SINCE WE KNOW THAT THE SAIL IS A RIGHT TRIANGLE

WE CAN USE THE PROPERTIES TO EXPLORE

DIFFERENT TYPES OF SAILS.

YOU HAVE PROBABLY SEEN SAILBOATS

WITH VERY TALL SAILS

AND SOME WITH SHORTER, MORE MODEST SAILS.

AN IMPORTANT STATISTIC USED WITH SAILBOATS

IS KNOWN AS THE ASPECT RATIO.

USING THIS RIGHT TRIANGLE, THE ASPECT RATIO IS

B SQUARED DIVIDED BY THE AREA OF THE TRIANGLE.

LET'S EXPLORE THE ASPECT RATIO ON THE NSPIRE.

RETURNING TO THE CONSTRUCTION

YOU PREVIOUSLY MADE, RECALL THAT YOU HAD FIXED

THE LENGTHS OF THE BASE AND THE VERTICAL LEG

IN ORDER TO ALLOW THE ANGLE TO VARY.

NOW WE WANT TO SWITCH THINGS AROUND.

WE WANT TO FIX THE ANGLE AT 90 DEGREES AND ALLOW

THE TWO LEGS OF THE RIGHT TRIANGLE TO VARY.

MOVE THE POINTER ABOVE THE MEASUREMENT FOR SIDE B.

PRESS CONTROL AND MENU

AND CHOOSE THE ATTRIBUTES OPTION.

USE THE DOWN ARROW TO HIGHLIGHT

THE LENGTH MEASUREMENT AND USE THE LEFT ARROW

TO CHANGE THE CLOSED LOCK TO AN OPEN LOCK.

PRESS ENTER.

REPEAT FOR THE OTHER LEG OF THE TRIANGLE.

PRESS ENTER TO CHANGE THE CLOSED LOCK TO AN OPEN LOCK.

NOW MAKE SURE YOU HAVE A 90 DEGREE ANGLE

AT THE CORNER OF THE TRIANGLE.

TRY AND GET YOUR ANGLE AS CLOSE TO 90 DEGREES

AS POSSIBLE. ONCE YOU HAVE DONE THIS,

HOVER OVER THE VALUE OF THE ANGLE MEASUREMENT

AND CHANGE ITS ATTRIBUTE FROM UNLOCKED TO LOCKED.

NOW WHEN YOU TRY TO MODIFY THIS TRIANGLE,

THE CORNER ANGLE REMAINS FIXED

WHILE THE TWO LEGS CAN VARY IN SIZE.

LET'S CREATE A NEW FORMULA TO TRACK THE ASPECT RATIO.

WE ALREADY HAVE MEASUREMENTS FOR B

AND THE AREA OF THE TRIANGLE,

SO PRESS MENU AND UNDER ACTIONS SELECT TEXT.

MOVE THE CURSOR TO A CLEAR AREA OF THE SCREEN

AND PRESS ENTER.

INPUT THE FORMULA B SQUARED OVER AREA AND PRESS ENTER.

NOW PRESS MENU AND UNDER ACTIONS SELECT CALCULATE.

HOVER OVER THE NEW FORMULA AND PRESS ENTER.

MOVE THE CURSOR TO LINK THE APPROPRIATE MEASURE.

PLACE THE MEASUREMENT NEXT TO THE FORMULA.

CHANGE THE HEIGHT OF THE VERTICAL LEG

AND MAKE A NOTE OF THE ASPECT RATIO.

NOTICE THAT A HIGH ASPECT RATIO CORRESPONDS TO

TALLER SAILBOATS WHILE A SHORTER ASPECT RATIO

CORRESPONDS TO A SHORTER SAILBOAT.

IN GENERAL, HIGHER ASPECT RATIOS

ARE ASSOCIATED WITH FASTER SAILBOATS.

IN FACT, SAILBOATS USED FOR RACING

HAVE VERY HIGH ASPECT RATIOS.

BUT THERE ARE LIMITS.

IF A SAILBOAT IS TOO TALL THEN ITS CENTER OF GRAVITY

IS ALSO HIGHER, MAKING THE BOAT MORE INCLINED

TO TIP OVER IN A VERY STRONG WIND.

LOOKING AT THE FORMULA FOR THE ASPECT RATIO

AND USING THE PYTHAGOREAN THEOREM

WE CAN REWRITE THE FORMULA TO LOOK LIKE THIS:

FOR SIMPLICITY, LET C = 1 AND LET'S REPLACE B WITH X.

WE THEN DERIVE THIS FUNCTION:

F OF X EQUALS 2X OVER THE SQUARE ROOT OF

THE QUANTITY 1 MINUS X SQUARED.

A GRAPH OF THE FUNCTION SHOWS THAT THE ASPECT RATIO

SHOWS A DRAMATIC INCREASE PAST A CERTAIN POINT.

IN OTHER WORDS, THE SAIL CANNOT EXCEED

A CERTAIN LENGTH BEYOND WHICH IT BECOMES

UNMANAGEABLE IN A STRONG WIND.

EXPLORE THE GRAPH OF THIS FUNCTION

USING THE GRAPHS AND GEOMETRY WINDOW.

TRY DIFFERENT VALUES FOR C.

IN ALL CASES YOU'LL SEE A SIMILARLY SHAPED GRAPH.

AS AN EXTENSION TO THIS TOPIC YOU'LL SEE THAT THE

TRIGONOMETRIC RATIO KNOWN AS THE TANGENT

IS NEARLY IDENTICAL TO THE ASPECT RATIO.

AND THE GRAPH OF THE TANGENT FUNCTION

HAS A SIMILAR SHAPE TO THE GRAPH

OF THE ASPECT RATIO FUNCTION.

SO SAILING NOT ONLY HAS A STRONG CONNECTION

TO RIGHT TRIANGLE GEOMETRY,

IT ALSO HAS A CLEAR CONNECTION

TO RIGHT TRIANGLE TRIGONOMETRY.