- WELCOME TO OUR VIDEO LESSON ON THE APPLICATIONS
OF TRIGONOMETRIC EQUATIONS
AND OUR GOAL IS TO SOLVE APPLICATION PROBLEMS.
SO HERE WE HAVE A FUNCTION
THAT MODELS THE AVERAGE MONTHLY TEMPERATURE OF THE WATER
IN A MOUNTAIN STREAM.
WHERE T OF X IS A TEMPERATURE IN FAHRENHEIT OF THE WATER
IN THE MONTH X WHERE X EQUALS ONE AS JANUARY,
X EQUALS TWO WOULD BE FEBRUARY AND SO ON.
SO WE HAVE TWO QUESTIONS HERE.
WHAT IS THE TEMPERATURE OF THE WATER IN OCTOBER?
WELL, October WOULD BE THE 10TH MONTH OF THE YEAR
SO IT'S PRETTY STRAIGHTFORWARD HERE,
WE HAVE TO REPLACE X WITH 10 AND EVALUATE THIS FUNCTION.
LET'S GO AHEAD AND DO THAT.
SO PI/6 x 10 - PI/2 AND THEN WE'LL ADD 53 TO THAT.
SO LET'S GO AHEAD AND GET OUR CALCULATOR OUT
AND WE'LL MAKE SURE WE'RE IN RADIAN MODE,
AND THE REASON I KNOW THAT IS PI/6 AND PI/2
SO WE'RE JUST GOING TO TYPE THIS IN
AND THEN SEE WHAT THE TEMPERATURE WOULD BE.
AND IT WAS 53 SO IT LOOKS LIKE IN OCTOBER
THE WATER TEMPERATURE WAS APPROXIMATELY
43.5 DEGREES FAHRENHEIT.
SO THAT'S A PRETTY STRAIGHTFORWARD QUESTION.
NOW THE NEXT ONE SAYS WHAT TWO MONTHS
ARE LIKELY TO GIVE A TEMPERATURE OF 62.4 DEGREES?
LET'S TAKE THIS ON THE NEXT SCREEN
AND SEE IF WE CAN SOLVE THAT.
SO NOW THEY'RE GIVING US THE TEMPERATURE OR T OF X
AND THEY WANT US TO FIND THE POSSIBLE VALUES OF X
OR IN WHICH MONTHS THE TEMPERATURE
WILL BE 62.4 DEGREES.
LET'S GO AHEAD AND SET THIS UP.
SO LET'S GO AHEAD AND ISOLATE THE SINE FUNCTION.
SO WE'LL SUBTRACT 53 ON BOTH SIDES,
THAT'LL GIVE US 11.2, NOW WE'LL DIVIDE BY 19,
THIS COMES OUT TO APPROXIMATELY .4947.
NOW WHAT WE CAN DO IS TAKE THE INVERSE SINE ON BOTH SIDES
SO ON THE LEFT, WE HAVE THE INVERSE SINE
OF .4947 = INVERSE SINE OF SINE OF THIS QUANTITY
WILL GIVE US JUST THIS QUANTITY.
LET'S GO TO THE CALCULATOR AND EVALUATE THIS.
INVERSE SINE .4947, THIS IS GOING TO GIVE US
AN ANGLE OF APPROXIMATELY .5175 RADIANS.
BUT REMEMBER THE SINE FUNCTION IS ALSO POSITIVE
IN THE SECOND QUADRANT
SO THERE'S GOING TO BE A SECOND ANGLE HERE
THAT HAS A SIGN FUNCTION VALUE OF .4947
SO EVEN THOUGH IT'S NOT IN THE RANGE OF INVERSE SINE
THE ORIGINAL EQUATION DOES NOT HAVE INVERSE SINE IN IT
SO WE ARE GOING TO HAVE TO FIND THAT OTHER ANGLE.
SO WE FOUND THIS ANGLE OVER HERE
BUT WE NEED TO FIND ANGLE IN THE SECOND QUADRANT
THAT HAS THE SAME REFERENCE ANGLE.
SO THIS ANGLE IS GOING TO BE PI - .5175
WHICH COMES OUT TO APPROXIMATELY 2.6241.
SO WE ACTUALLY HAVE TWO EQUATIONS THAT WE HAVE TO SOLVE.
LET'S GO AHEAD AND TAKE THIS INFORMATION
ON TO THE NEXT SCREEN.
WE'RE GOING TO SET THE RIGHT SIDE EQUAL TO THE .5175
AND THEN WE'LL TAKE THE RIGHT SIDE
AND SET IT EQUAL TO 2.6241 AS WELL.
AGAIN, WE HAVE TWO DIFFERENT ANGLES
THAT HAVE THE SINE FUNCTION VALUE
THAT WE WERE LOOKING FOR,
WE HAVE THIS AND WE ALSO HAVE THIS ANGLE.
SO IN ORDER TO SOLVE THIS -- WE'LL ADD PI/2 TO BOTH SIDES.
LET'S GO AHEAD AND DO THE SAME THING ON THIS SIDE.
THIS WILL BE APPROXIMATELY 2.0883 = PI/6X
AND ON THIS SIDE WE'RE GOING TO HAVE 4.1949 = PI/6X.
NOW WE'LL MULTIPLY BY THE RECIPROCAL
OR MULTIPLY IT BY 6/PI
SO THIS SIMPLIFIES NICELY
AND WE'RE LEFT WITH X
IS APPROXIMATELY EQUAL TO 3.9884 AND OVER HERE,
WE'LL HAVE X IS APPROXIMATELY 8.0117.
SO THIS IS VERY CLOSE TO FOUR WHICH WOULD BE APRIL
AND THIS IS VERY CLOSE TO EIGHT WHICH WOULD BE AUGUST.
SO THESE WOULD BE THE TWO MONTHS WHERE WE WOULD EXPECT
THE WATER TEMPERATURE TO BE 62.4 DEGREES.
SO IT WAS A LITTLE INVOLVED SO WE HAVE TO BE SURE
WHEN WE USE INVERSE SINE THAT EVEN THOUGH THE SECOND VALUE
IS NOT IN THE RANGE OF THIS
THIS INVERSE FUNCTION WAS NOT IN THE ORIGINAL EQUATION
SO WE DO HAVE TO FIND THAT SECOND ANGLE.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE QUESTION.
RAUL'S HEATING OIL IS VERY SEASONAL
WITH SALES IN THE WINTER FAR EXCEEDING SALES IN THE SUMMER.
MONTHLY SALES CAN BE MODELED BY THIS FUNCTION
WHERE S OF X IS THE AVERAGE MONTHLY SALES
IN THE MONTH X WHERE X EQUALS ONE AS JANUARY AND SO ON.
SO THE FIRST QUESTION WHAT IS THE MONTHLY
AVERAGE SALES IN JULY?
WELL JULY IS THE SEVENTH MONTH SO WE HAVE X = 7
SO JUST AS BEFORE WE'LL JUST REPLACE X WITH 7.
WE JUST DID ONE LIKE THIS SO I WON'T TAKE THE TIME
TO TYPE IT INTO THE CALCULATOR;
IT COMES OUT TO APPROXIMATELY $3554.52.
OKAY, FOR THE NEXT PART FOR WHAT MONTHS OF THIS YEAR
ARE SALES LESS THAN $4000?
I THINK WHAT WE'LL DO THIS TIME IS, IS LOOK AT THIS GRAPHICALLY.
SO WE'LL GRAPH THIS FUNCTION AND DETERMINE
FOR WHICH X VALUES IT'S BELOW THE LINE Y = 4000.
SO LET'S GET OUR GRAPHING CALCULATORS OUT,
MAKE SURE WE'RE IN RADIAN MODE,
NEXT, LET'S SET UP THE WINDOW
SO WE GET A NICE VIEW AND I'VE ALREADY SET IT UP NICELY
X GOES FROM -1 TO 13 BY 1.
REMEMBER, X IS THE MONTH,
Y IS THE DOLLAR VALUE OF SALES
AND I SET IT UP FROM -200 TO 8000.
I LIKE TO SHOW NEGATIVE VALUES HERE
SO WE CAN SEE THE ORIGIN ON THE COORDINATE PLANE
AND WE'LL GO BY 500S.
NEXT, I ALREADY TYPED IN THE FUNCTION INTO Y1
AND THEN IN Y2 WE'RE GOING TO TYPE IN $4000
BECAUSE WE WANT TO KNOW IN WHICH MONTHS
THE SALES ARE LESS THAN $4000.
SO WE WANT TO SEE WHEN Y1 IS BELOW Y2.
WE PRESS GRAPH;
YOU GET A NICE VISUAL OF HOW THE SALES ARE HIGH
IN THE BEGINNING OF THE YEAR THEY GO DOWN IN THE SUMMER
AND THEN BACK UP IN THE FALL AND WINTER.
THE SALES WOULD BE BELOW $4000 IN THIS INTERVAL HERE.
SO WE NEED TO DETERMINE WHAT THE X VALUE IS HERE
AND WHAT THE X VALUE IS HERE.
IF WE PRESS SECOND TRACE OPTION FIVE FOR INTERSECTION
AND SCROLL OVER TO THE LEFT,
IT ONLY FINDS ONE POINT OF INTERSECTION AT A TIME.
LET'S PRESS ENTER THREE TIMES.
SO X IS APPROXIMATELY 4.94.
LET'S GO AHEAD AND RECORD THAT.
SO IF X IS GREATER THAN 4.94
AND LET'S GO AHEAD AND FIND THE SECOND POINT OF INTERSECTION.
SCROLL OVER TO THE RIGHT PRESS ENTER THREE TIMES.
THIS VALUE IS APPROXIMATELY X = 8.05.
SO WHEN X IS BETWEEN 4.94 AND 8.05 THE SALES ARE BELOW $4000.
WELL, THIS IS VERY CLOSE TO THE FIFTH MONTH
WHICH WOULD BE MAY AND THE EIGHTH MONTH WOULD BE AUGUST.
THE SALES WOULD BE LESS THAN $4000.
NOW ONE THING WE HAVE TO BE CAREFUL ABOUT HERE,
SOME TEXTBOOKS ARE VERY TECHNICAL.
IF WE SAY X EQUALS EIGHT IS AUGUST
THEN THAT MEANS FROM 7 TO 8
WOULD BE AUGUST ANYTHING LARGER
THAN 8 WOULD ACTUALLY FALL INTO THE NEXT MONTH OR SEPTEMBER.
YOU CAN SEE I'M ROUNDING TO THE NEAREST MONTH.
SO THIS IS CLOSE ENOUGH TO FIVE FOR ME
AND THIS IS CLOSE ENOUGH TO EIGHT
THEREFORE WE'LL SAY BETWEEN MAY AND AUGUST.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU AND HAVE A GOOD DAY.