WELCOME TO A VIDEO ON THE LAW OF COSINES.
THE GOALS OF THE VIDEO ARE TO DERIVE THE LAW OF COSINES
AND ALSO TO SOLVE TRIANGLES USING THE LAW OF COSINES.
SO JUST TO REVIEW,
A TRIANGLE THAT IS NOT A RIGHT TRIANGLE
IS CALLED AN OBLIQUE TRIANGLE.
AND UNTIL NOW WE'VE ONLY BEEN ABLE TO SOLVE RIGHT TRIANGLES.
HOWEVER, THE LAW OF COSINES DOES ALLOW US
TO SOLVE CERTAIN OBLIQUE TRIANGLES.
SO THE REQUIREMENTS FOR SOLVING OBLIQUE TRIANGLES
USING THE LAW OF COSINES IS SIDE, ANGLE, SIDE,
WHERE TWO SIDES AND AN INCLUDED ANGLE ARE KNOWN,
OR SIDE, SIDE, SIDE WHERE ALL THREE SIDES ARE KNOWN.
SO HERE'S THE LAW OF COSINES FOR ANY TRIANGLE,
"A," B, C AND REMEMBER THAT SIDE "A" IS OPPOSITE ANGLE "A"
SIDE B IS OPPOSITE ANGLE B, AND SIDE C IS OPPOSITE ANGLE C.
SO IF WE TAKE A LOOK AT THESE THREE FORMULAS,
THE MOST IMPORTANT PART TO REMEMBER
IS WHATEVER ANGLES WE'RE USING, THE SIDE OPPOSITE THAT ANGLE
WOULD BE THE SIDE ON THE OTHER SIDE OF THE EQUAL SIGN.
LET'S GO AHEAD
AND TAKE A LOOK AT ONE WAY TO PROVE THIS THEOREM.
IF WE TAKE ANY TRIANGLE AND CONSTRUCT AN ALTITUDE
FROM LET'S SAY VERTEX B,
YOU WOULD HAVE 2 RIGHT TRIANGLES,
ONE HERE AND ONE HERE.
NEXT IF THIS ENTIRE LENGTH IS B, WE CAN LET THIS LENGTH BE X,
THEREFORE THIS LENGTH WOULD BE B - X.
APPLYING THE PYTHAGOREAN THEOREM TO THIS SMALL RIGHT TRIANGLE
WE WOULD HAVE C SQUARED = Y SQUARED + (B - X SQUARED).
NOW WE CAN REPLACE X SQUARED + Y SQUARED WITH "A" SQUARED
FROM THE LARGER RIGHT TRIANGLE.
NEXT IF WE REFERENCE ANGLE C,
THE COSINE OF ANGLE C WOULD BE EQUAL TO X/A SOLVING THIS FOR X,
WE CAN REPLACE X WITH "A" COSINE C.
AND THEN WE HAVE THE LAW OF COSINES.
AND WE CAN SHOW THE OTHER FORMS BY CONSTRUCTING ALTITUDES
FROM THE OTHER TWO VERTICES.
SO LET'S TAKE A LOOK AT A COUPLE PROBLEMS.
AGAIN IT'S NOT IMPORTANT THAT THIS IS TO SCALE,
WHAT'S IMPORTANT IS WE SHOW THE RELATIONSHIPS PROPERLY.
SO WE KNOW ANGLE B IS 73.1 DEGREES,
"A" IS 24.2 FEET, AND C IS 43.7 FEET.
SO THE FIRST THING WE SHOULD NOTICE IS
THAT ACROSS FROM ANGLE B IS SIDE B.
SO IF WE USE ANGLE B,
THE SIDE B WILL BE ON THE LEFT SIDE OF THE EQUAL SIGN
SO WE'LL BE LOOKING AT THIS FORM OF THE LAW OF COSINES.
LET'S GO AHEAD AND SET THIS UP.
B SQUARED MUST EQUAL "A" SQUARED + C SQUARED - 2,
x "A" x C, x THE COSINE OF B, WHICH IS 73.1 DEGREES.
TO SAVE TIME I'VE ALREADY FOUND THE VALUE OF THIS RIGHT SIDE,
B SQUARED IS APPROXIMATELY EQUAL TO 1,880.47.
AGAIN I EVALUATED THIS ON THE CALCULATOR.
WE DON'T HAVE TIME TO SHOW IT IN THIS VIDEO.
NOW WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION
SO WE HAVE B IS APPROXIMATELY EQUAL TO 43.4.
LET'S GO AHEAD AND LABEL THAT IN THE TRIANGLE.
NOW WE HAVE THE LENGTHS OF ALL OF THE SIDES
AND WE HAVE THE MEASURE OF ONE OF THE ANGLES.
NOW WE CAN USE THE LAW OF SINES
TO DETERMINE ONE OF THE MISSING ANGLES.
LET'S GO AHEAD AND SET THAT UP.
LET'S GO AHEAD AND USE ANGLE B AND SO THE SINE OF 73.1 DEGREES
DIVIDED BY THE OPPOSITE SIDE WHICH IS 43.4 FEET
MUST EQUAL THE SINE OF--
LET'S GO AHEAD AND TRY TO FIND THE MEASURE OF ANGLE "A."
SINE OF ANGLE "A" DIVIDED BY 24.2 FEET.
NOW WE'LL CROSS MULTIPLY 43.4 OF SINE "A"
MUST EQUAL 24.2 x SINE 73.1 DEGREES.
WE'LL DIVIDE BOTH SIDES BY 43.4,
ON THE LEFT SIDE WE'LL HAVE SINE "A,"
LET'S GO AHEAD AND BRING IT DOWN HERE.
IT'S APPROXIMATELY EQUAL
TO THIS QUOTIENT WHICH IS APPROXIMATELY 0.5335.
SO NOW WE CAN TAKE THE INVERSE SINE OF BOTH SIDES
AND "A" IS GOING TO BE
APPROXIMATELY EQUAL TO THE INVERSE SINE, 0.5335,
SO ANGLE "A" IS APPROXIMATELY EQUAL TO 32.2 DEGREES.
LET'S GO AHEAD AND LABEL THAT, SO THIS ANGLE IS 32.2 DEGREES.
WE KNOW THE SUM MUST BE 180 DEGREES
SO 180 - 32.2 - 73.1 WILL GIVE US 74.7 DEGREES FOR ANGLE C.
SO OFTEN AFTER USING THE LAW OF COSINES
WE WILL USE THE LAW OF SINES TO FINISH SOLVING THE TRIANGLE.
LET'S TAKE A LOOK AT ONE MORE.
HERE WE HAVE THE LENGTH OF THREE SIDES OF THE TRIANGLE.
SO SIDE "A" IS 25.4 CENTIMETERS,
SIDE B IS 42.8 CENTIMETERS AND SIDE C IS 59.3 CENTIMETERS.
SO WE HAVE TO DECIDE WHICH ANGLE WE WANT TO FIND FIRST
AND IT'S RECOMMENDED THAT WE FIND THE LARGEST ANGLE FIRST
TO AVOID ANY POSSIBLE COMPLICATIONS
WITH THE AMBIGUOUS CASE OF THE LAW OF SINES.
SO REMEMBER THE LARGEST ANGLE IS OPPOSITE THE LARGEST SIDE.
SO LET'S SEE IF WE CAN FIND THE MEASURE OF ANGLE C FIRST.
IF WE WANT TO FIND ANGLE C,
WE'RE GOING TO HAVE TO USE THIS FORM OF THE LAW OF COSINES.
SO WE'LL HAVE C SQUARED MUST EQUAL "A" SQUARED + B SQUARED
- 2 x "A" x B x COSINE C.
SO NOTICE IT LOOKS DIFFERENT THAN LAST TIME
BECAUSE THE UNKNOWN IS NOW ON THE RIGHT SIDE.
WE NEED TO GO AHEAD AND SIMPLIFY ALL OF THIS.
I WILL SHOW A LITTLE BIT OF WORK HERE
BUT YOU PROBABLY WANT TO CHECK IT ON YOUR CALCULATOR.
COMBINE LIKE TERMS, SUBTRACT 2,477 FROM BOTH SIDES,
DIVIDE BY -2,174 ON BOTH SIDES
SO WE'LL HAVE -0.3830 = COSINE C.
NOW IF WE TAKE INVERSE COSINE OF BOTH SIDES,
WE'LL HAVE C IS APPROXIMATELY EQUAL
TO INVERSE COSINE OF -0.3830.
SO ANGLE C IS APPROXIMATELY 112.5 DEGREES.
LET'S GO AHEAD AND LABEL THAT.
NOW TO FIND A SECOND ANGLE IN THIS TRIANGLE,
WE'LL USE THE LAW OF SINES.
SO WE'LL USE ANGLE C AND SIDE C TO START.
SINE OF 112.5 DEGREES DIVIDED BY 59.3 MUST EQUAL,
LET'S GO AHEAD AND FIND THE MEASURE OF ANGLE B.
SO THE SINE OF B DIVIDED BY 42.8.
NOW WE'LL CROSS MULTIPLY,
DIVIDE BY 59.3 SO SINE B IS APPROXIMATELY 0.6668.
SO IF WE TAKE THE INVERSE SINE OF BOTH SIDES,
SO ANGLE B IS APPROXIMATELY 41.8 DEGREES.
NOW WE CAN DETERMINE THIS LAST ANGLE
SINCE WE KNOW THE SUM MUST BE 180 DEGREES,
SO ANGLE "A" MEASURES APPROXIMATELY 25.7 DEGREES.
OKAY. THAT SHOULD DO IT FOR THE LAW OF COSINES.
THERE IS ANOTHER VIDEO THAT DEALS WITH APPLICATIONS
INVOLVING LAW OF COSINES.
THANK YOU FOR WATCHING AND HAVE A GOOD DAY.