Verifying Identities:  Sum, Difference, Double, and Half Angle Identities

Verifying Identities: Sum, Difference, Double, and Half Angle Identities


– 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.  

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