[ MUSIC ]
- WELCOME TO A LESSON ON THE LAW OF SINES.
THE GOALS OF THIS VIDEO ARE TO DERIVE
THE LAW OF SINES AND ALSO, TO SOLVE TRIANGLES
USING THE LAW OF SINES.
BUT FIRST, A TRIANGLE THAT IS NOT A RIGHT TRIANGLE
IS CALLED AN OBLIQUE TRIANGLE AS WE SEE HERE.
UNTIL NOW, WE'VE ONLY BEEN ABLE TO SOLVE RIGHT TRIANGLES,
HOWEVER THE LAW OF SINES WILL ALLOW US TO SOLVE
OBLIQUE TRIANGLES AS WELL AS SOLVE
NEW APPLICATION PROBLEMS.
SO HERE ARE THE REQUIREMENTS FOR SOLVING OBLIQUE TRIANGLES
USING THE LAW OF SINES.
WE MUST BE GIVEN ANGLE-ANGLE- SIDE OR ANGLE-SIDE-ANGLE
MEANING ONE SIDE AND TWO ANGLES ARE KNOWN OR SIDE-SIDE-ANGLE
WHERE TWO SIDES AND ONE ANGLE NOT INCLUDED BETWEEN
THE TWO SIDES ARE KNOWN.
NOW SIDE-SIDE-ANGLE TENDS TO BE A LITTLE MORE INVOLVED
BECAUSE IT CAN LEAD TO MORE THAN ONE POSSIBLE TRIANGLE
SO THIS CASE WILL BE COVERED IN A DIFFERENT VIDEO
AND WE SHOULD MENTION THAT OBLIQUE TRIANGLES
WITH SIDE-ANGLE-SIDE AND SIDE-SIDE-SIDE
CAN ALSO BE SOLVED USING SOMETHING CALLED
THE LAW OF COSINES WHICH IS COVERED
IN A DIFFERENT VIDEO AS WELL.
SO LET'S GO AHEAD AND STATE THE LAW OF SINES.
FOR ANY TRIANGLE "A," B, C THE RATIO OF THE SINE
OF AN ANGLE TO THE OPPOSITE SIDE OF THAT ANGLE IS CONSTANT
AND IT CAN BE EXPRESSED IN TWO DIFFERENT WAYS.
WE EITHER HAVE THE LENGTH OF THE SIDES IN THE NUMERATOR
AND THE SINES OF ANGLES IN THE DENOMINATOR
OR WE CAN STATE IT USING THE RECIPROCALS
AND I LIKE TO USE THIS FORM.
THE SINE OF ANGLE "A" DIVIDED BY THE LENGTH
OF SIDE "A" MUST EQUAL THE SINE OF ANGLE B
DIVIDED BY THE LENGTH OF SIDE B, WHICH MUST EQUAL
THE SINE OF ANGLE C DIVIDED BY THE LENGTH OF SIDE C.
SO WHAT THIS IS SAYING IS THAT THESE RATIOS
WILL BE CONSTANT IN ANY TRIANGLE
AND WHAT'S GOING TO HAPPEN IS WE'RE GOING TO USE TWO
OF THESE RATIOS TO SET UP A PROPORTION WHERE ONE
OF THESE VALUES WILL BE UNKNOWN, AND THEN WE CAN USE THE FACT
THAT CROSS PRODUCTS ARE EQUAL IN A PROPORTION
AND SOLVE FOR THE UNKNOWN VALUE.
BUT BEFORE WE DO THAT LETS TAKE A LOOK
AT AN ANIMATION.
UP HERE IN THIS TABLE, WE HAVE THE RATIOS OF THE SINES
OF THE ANGLES DIVIDED BY THE LENGTH
OF THE OPPOSITE SIDES AND WHAT WE'LL NOTICE
IS WHEN WE CHANGE THE TRIANGLE BY CHANGING ANY ANGLE
OR THE LENGTH OF ANY SIDE NOTICE THAT THESE RATIOS
REMAIN CONSTANT.
THIS IS ILLUSTRATING THE PROPERTY
OF THE LAW OF SINES.
SO REGARDLESS OF THE TRIANGLE, IF WE TAKE THE SINE OF THE ANGLE
AND DIVIDE BY THE LENGTH OF THE OPPOSITE SIDE
THAT RATIO REMAINS CONSTANT.
LET'S GO AHEAD AND SEE IF WE CAN ACTUALLY DERIVE
THE LAW OF SINES.
IF WE HAVE ANY TRIANGLE "A," B, C AND CONSTRUCT AN ALTITUDE
FROM LETS SAY VERTEX C TO THE OPPOSITE SIDE
NOTICE WHAT HAPPENS IS WE FORM TWO RIGHT TRIANGLES
A SMALLER RIGHT TRIANGLE AND A LARGER RIGHT TRIANGLE.
SO WE COULD SAY THE SINE OF ANGLE "A" IS EQUAL TO H/B
IN THIS SMALL RIGHT TRIANGLE AND WE COULD ALSO STATE
THAT THE SINE OF ANGLE B IS EQUAL TO H/A
IN THE LARGER RIGHT TRIANGLE.
SO IF WE HAVE THESE TWO EQUATIONS
AND WE SOLVE THEM FOR H, WE WOULD HAVE H = B SINE
"A" AND H = "A" SINE B.
SINCE THEY'RE BOTH EQUAL TO H WE COULD STATE THAT B SINE "A"
IS EQUAL TO "A" SINE B AND NOW IF WE DIVIDE BOTH SIDES
OF THIS EQUATION BY "AB" NOTICE ON THE LEFT
THE B'S SIMPLIFY OUT AND NOTICE ON THE RIGHT
THE "A'S" SIMPLIFY OUT.
SO WE'RE LEFT WITH SINE OF ANGLE "A" DIVIDED
BY THE LENGTH OF SIDE "A" MUST EQUAL THE SINE OF ANGLE B
DIVIDED BY THE LENGTH OF B.
TO GET THE 3rd RATIO STATED IN THE LAW OF SINES
WE COULD CONSTRUCT AN ALTITUDE FROM VERTEX "A" AND SHOW
THIS IS ALSO EQUAL TO SINE OF ANGLE C DIVIDED
BY THE LENGTH OF SIDE C.
OKAY. LET'S GO AHEAD AND TAKE A LOOK AT A PROBLEM.
WE WANT TO SOLVE THE TRIANGLE WHERE ANGLE "A" IS 48 DEGREES,
ANGLE B IS 54 DEGREES, AND SIDE "A"
HAS LENGTH 12.5 FEET.
NOW IT'S NOT IMPORTANT THAT OUR TRIANGLE
IS TO SCALE BUT WHAT IS IMPORTANT
IS THAT IF WE CALL THIS ANGLE "A" THIS MUST BE SIDE "A."
IF WE CALL THIS ANGLE B THE OPPOSITE SIDE MUST BE SIDE B
AND IF THIS IS ANGLE C THIS MUST BE SIDE C.
LET'S GO AHEAD AND RECORD THE GIVEN INFORMATION.
ANGLE "A" IS 48 DEGREES.
ANGLE B IS 54 DEGREES AND SIDE "A" IS 12.5 FEET.
OKAY. BEFORE WE GET TOO EXCITED AND TRY TO USE THE LAW OF SINES,
WE SHOULD TAKE ANOTHER LOOK AT THIS TRIANGLE.
WE KNOW THE SUM OF THE INTERIOR ANGLES MUST EQUAL 180 DEGREES.
SO WE CAN GO AHEAD AND FIND THE MEASURE OF ANGLE C
SINCE WE KNOW THAT C + 48 DEGREES + 54 DEGREES
MUST EQUAL 180 DEGREES.
SO THE MEASURE OF ANGLE C IS EQUAL TO 78 DEGREES.
SO NOW WHAT WE'RE LEFT TO FIND ARE THE LENGTHS OF SIDE B
AND SIDE C.
THIS WILL REQUIRE THE LAW OF SINES.
AGAIN, WHAT WE'RE GOING TO DO IS PICK TWO OF THESE RATIOS
WHERE THREE OF THE VALUES ARE KNOWN AND THEN WE'LL SOLVE
FOR THE REMAINING UNKNOWN VALUE.
SO WE DO HAVE TO HAVE THE MEASURE OF ONE ANGLE
AND IT'S OPPOSITE SIDE TO START WITH AND SO WE'LL USE
ANGLE "A" AND SIDE "A" TO GET US STARTED.
SO THE SINE OF 48 DEGREES IS TO 12.5 FEET AS--
NOW TO COMPLETE OUR PROPORTION LET'S GO AHEAD AND USE ANGLE B.
SO WE'LL HAVE THE SINE OF 54 DEGREES TO SIDE B.
SO AGAIN, NOTICE WE HAVE ONE UNKNOWN VALUE
SO NOW WE CAN CROSS MULTIPLY.
SO WE HAVE B x SINE 48 MUST EQUAL 12.5 x SINE 54.
SO NOW WE CAN SOLVE FOR B BY DIVIDING BOTH SIDES
BY SINE 48 DEGREES.
SO NOW WE'LL HAVE B IS APPROXIMATELY EQUAL
TO THIS QUOTIENT.
LET'S GO AHEAD AND GET OUR CALCULATORS OUT,
MAKE SURE THAT WE ARE IN DEGREE MODE AND OUR NUMERATOR
WILL BE 12.5 x SINE 54 DEGREES DIVIDED BY SINE 48 DEGREES
AND SO IT LOOKS LIKE IT'S ABOUT 13.6 FEET FOR SIDE B.
NOW WE STILL HAVE TO FIND THE LENGTH OF SIDE C.
WE CANNOT USE THE PYTHAGOREAN THEOREM
BECAUSE IT'S NOT A RIGHT TRIANGLE
BUT WE CAN USE THE LAW OF SINES AGAIN.
TO GET US STARTED I WILL USE THE RATIO OF THE SINE
OF ANGLE "A" DIVIDED BY THE LENGTH OF SIDE "A"
AND NOW WE'LL USE ANGLE C SO THE SINE OF 78 DEGREES
DIVIDED BY THE LENGTH OF SIDE C.
SO AGAIN, THIS PROPORTION IS STATING THAT THE SINE
OF 48 DEGREES IS TO 12.5 AS THE SINE OF 78 DEGREES
IS TO THE LENGTH OF SIDE C.
WE'LL GO AHEAD AND CROSS MULTIPLY AGAIN.
SO WE HAVE C SINE 48 DEGREES MUST EQUAL 12.5 x THE SINE
OF 78 DEGREES.
TO SOLVE FOR THE LENGTH OF C WE'LL DIVIDE BY SINE 48 DEGREES.
NOW WE'LL GO BACK TO THE CALCULATOR
AND FIND THE VALUE OF THIS QUOTIENT.
12.5 SINE 78 DEGREES DIVIDED BY THE SINE 48 DEGREES
AND C IS APPROXIMATELY 16.5 FEET.
NOW ONE LAST CHECK TO MAKE SURE WE DIDN'T MAKE A CALCULATION
ERROR OR MISPLACE A DECIMAL.
THE LONGEST SIDE SHOULD BE OPPOSITE THE LARGEST ANGLE.
SO HERE'S OUR LARGEST ANGLE OF 78 DEGREES.
16.5 FEET IS THE LONGEST SIDE AND THE SHORTEST SIDE OF 12.5
SHOULD BE OPPOSITE THE SMALLEST ANGLE,
WHICH IT IS.
SO THIS IS A NICE WAY TO DOUBLE CHECK OUR WORK
ONE LAST TIME TO MAKE SURE WE DIDN'T MAKE A MISTAKE
ON THE CALCULATOR TO SOLVE OUR RIGHT TRIANGLE.
OKAY. SO THERE ARE A COUPLE MORE VIDEOS ON THE LAW OF SINES.
ONE MORE THAT DEALS WITH THE AMBIGUOUS CASE
AND AT LEAST ONE OTHER THAT WILL DEAL
WITH SOLVING APPLICATION PROBLEMS.
I HOPE YOU FOUND THIS INTRODUCTORY VIDEO HELPFUL.
THANK YOU AND HAVE A GOOD DAY.
[ MUSIC ]