- WELCOME TO ANOTHER VIDEO
ON VERIFYING TRIGONOMETRIC IDENTITIES.
THIS VIDEO WILL FOCUS ON THE SUM, DIFFERENCE,
DOUBLE AND HALF ANGLE IDENTITIES.
SO HERE'S A LIST OF ALL THE IDENTITIES
THAT WE'LL BE WORKING WITH IN THIS VIDEO.
ONE OF THE MOST CHALLENGING PARTS ABOUT VERIFYING IDENTITIES
USING THESE IS DECIDING WHICH ONE TO USE
AND SO SOMETIMES IT DOES TAKE A LOT OF PRACTICE
TO GET GOOD AT VERIFYING IDENTITIES
SO HOPEFULLY YOU'LL KEEP A POSITIVE ATTITUDE
AND STICK WITH IT.
SO LET'S GO AHEAD AND GET STARTED.
WE HAVE A CHOICE HERE TO EITHER USE THE DOUBLE ANGLE IDENTITY
OR WE COULD MULTIPLY THIS OUT ON THE LEFT.
I'M GOING TO GO AHEAD AND MULTIPLY THE LEFT SIDE OUT.
SO WE'LL FOIL THIS.
WHEN WE MULTIPLY THIS OUT WE'LL HAVE SINE "A" x SINE "A"
OR SINE SQUARED "A".
NEXT, WE'LL HAVE A SINE "A" x COSINE "A",
AND THEN WE HAVE A COSINE "A" x SINE "A."
SO WE HAVE TWO LIKE TERMS SO WE'LL HAVE
+ 2 SINE "A" COSINE "A" + COSINE SQUARED "A."
THE NEXT THING WE SHOULD NOTICE
IS SINE SQUARED "A" + COSINE SQUARED "A" IS EQUAL TO 1.
SO WE'LL HAVE 2 SINE "A" COSINE "A" + 1 EQUALS SINE 2A + 1
AND NOW BELIEVE IT OR NOT WE HAVE IT.
REMEMBER THE IDENTITY SINE 2A IS EQUAL TO 2 SINE "A" COSINE "A"
SO WE CAN REPLACE THIS WITH SINE OF 2A.
SO IT IS IMPORTANT THAT YOU ARE PRETTY COMFORTABLE
WITH THESE IDENTITIES BECAUSE IT WILL HELP YOU RECOGNIZE
WHEN TO PERFORM THE CORRECT SUBSTITUTIONS.
LET'S GO AHEAD AND TRY ANOTHER.
HERE ON THE RIGHT THERE'S NOT MUCH WE CAN DO
WITH COSINE X - SINE X
BUT NOTICE ON THE LEFT WE HAVE COSINE OF X + PI/4
WE CAN EXPAND THIS USING THE SUM IDENTITY FOR COSINE.
SO THE LEFT SIDE CAN BE REWRITTEN
AS THE COSINE OF "A" x COSINE B.
SO COSINE OF X x COSINE OF PI/4.
SINCE WE HAVE A SUM, WE'LL USE A DIFFERENCE ON THE LEFT.
SINE "A" OR SINE X x SINE B OR SINE PI/4.
NEXT, WE CAN EVALUATE COSINE PI/4 AND SINE PI/4.
IF YOU DON'T HAVE YOUR UNIT CIRCLE HANDY
THE SINE OF PI/4 AND THE COSINE OF PI/4
ARE BOTH EQUAL TO THE SQUARE ROOT OF 2/2
AND NOW WE'RE MAKING VERY GOOD PROGRESS.
YOU CAN SEE THAT THEY'RE ALMOST THE SAME NOW.
BOTH OF THESE TERMS HAVE A COMMON FACTOR
OF SQUARE ROOT 2/2
SO WE CAN FACTOR THAT OUT AND I BELIEVE
THEY'RE GOING TO MATCH NOW AND THEY DO
SO WE'RE DONE AND LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE.
NOW THIS PROBLEM CAN LOOK INTIMIDATING ESPECIALLY
IF YOU'RE NOT FAMILIAR WITH YOUR IDENTITIES.
SO IT IS IMPORTANT THAT YOU HAVE A GOOD LIST OF THEM
AND YOU KNOW HOW TO USE THEM,
AND ON THIS PROBLEM I'M GOING TO WORK FROM BOTH SIDES.
SO I CAN REPLACE SINE OF 2A WITH 2 SINE "A" COSINE "A"
AND ON THE RIGHT SIDE
I'M GOING TO USE THE HALF ANGLE IDENTITIES.
NOTICE THAT THIS IS COSINE SQUARED A/2
SO THAT'S GOING TO ELIMINATE THE SQUARE ROOT HERE.
SO COSINE SQUARED A/2 IS GOING TO EQUAL 1 + COSINE A/2.
AGAIN, SINCE THIS IS SQUARED
IT'S UNDOING THAT SQUARE ROOT - SINE SQUARED A/2
WE'LL HAVE 1 - COSINE A/2.
NEXT, ON THE LEFT SIDE WE HAVE A COMMON FACTOR OF 2
AS WELL AS SINE "A" SO ON THE LEFT
WE'RE LEFT WITH COSINE "A."
NOW ON THE RIGHT NOTICE WE HAVE A COMMON DENOMINATOR
SO LET'S GO AHEAD AND ADD THESE FRACTIONS.
NOW WE DO HAVE TO BE CAREFUL WHEN WE DO THIS.
WE'RE SUBTRACTING THIS ENTIRE QUANTITY.
SO LET'S PUT OUR PARENTHESES IN PLACE.
THE DENOMINATOR IS 2.
NEXT WE HAVE 1 - 1 THAT'S 0
AND THEN WE HAVE COSINE "A" - A -COSINE "A"
SO THAT WILL GIVE US 2 COSINE "A"
AND BELIEVE IT OR NOT IN JUST A COUPLE OF QUICK STEPS
WE HAVE VERIFIED THIS IDENTITY.
THESE TWO SIMPLIFY NICELY
AND SO WE HAVE COSINE "A" = COSINE "A."
AGAIN, BECAUSE I PROVIDED THE CORRECT IDENTITIES
FOR EACH OF THESE PROBLEMS
IT MAY SEEM A LOT EASIER THEN WHEN YOU OPEN A TEXTBOOK
AND START WORKING SOME PROBLEMS.
SO YOU JUST HAVE TO BE PATIENT WITH IT AND TRY DIFFERENT THINGS
AND BECOME FAMILIAR WITH THOSE IDENTITIES.
AGAIN, THANK YOU FOR WATCHING AND HAVE A GOOD DAY.