- WELCOME TO A VIDEO
ON VERIFYING TRIGONOMETRIC IDENTITIES.
LET'S GO AHEAD AND GET STARTED.
HERE ARE THE GUIDELINES FOR VERIFYING IDENTITIES.
NUMBER ONE, YOU SHOULD MEMORIZE
THE FUNDAMENTAL IDENTITIES.
THIS WILL HELP YOU TO IDENTIFY
HOW TO APPROACH THE VERIFICATIONS.
NUMBER TWO,
REWRITE THE MORE COMPLICATED SIDE INTO A SIMPLER FORM.
NUMBER THREE, IT MAY BE HELPFUL TO CONVERT EVERYTHING
IN TERMS OF SINE AND COSINE.
NUMBER FOUR, ANY FACTORING OR ALGEBRAIC OPERATION
SHOULD BE PERFORMED,
FOR EXAMPLE, ADDING, SUBTRACTING, MULTIPLYING.
AND NUMBER FIVE,
IF AN EXPRESSION CONTAINS A SUM OR DIFFERENCE,
MULTIPLY BOTH THE NUMERATOR AND DENOMINATOR
BY THE CONJUGATE.
NOW AGAIN, THESE ARE JUST GUIDELINES,
BUT THE BEST WAY TO GET GOOD AT VERIFYING IDENTITIES
IS TO PRACTICE.
AND THINK OF THIS IN TERMS OF A PUZZLE
THAT YOU'RE TRYING TO SOLVE,
AND HOPEFULLY YOU'LL HAVE SOME FUN ALONG THE WAY.
SO HERE ARE A COUPLE OF IDENTITIES WE WANT TO VERIFY.
THIS FIRST ONE, IF YOU TAKE A LOOK AT BOTH SIDES,
THERE'S REALLY ONLY ONE SIDE WE CAN WORK WITH.
THERE'S A PRODUCT HERE THAT WE CAN WORK WITH.
WE CAN'T REALLY DO MUCH WITH SINE THETA.
AND THE NEXT THING IS WE CAN REWRITE TANGENT THETA
AS SINE THETA DIVIDED BY COSINE THETA.
AND IT'S PROBABLY HELPFUL IF WE WRITE COSINE THETA / 1,
AND THEN IT BECOMES VERY OBVIOUS
HOW THIS CAN BE SIMPLIFIED.
WE HAVE A COMMON FACTOR OF COSINE THETA,
AND NOW WE'VE VERIFIED OUR IDENTITY.
SINE THETA = SINE THETA.
NOW IT IS IMPORTANT WHEN WE VERIFY IDENTITIES
THAT WE DO SHOW THOROUGH WORK,
SO THAT IF SOMEONE WAS TO LOOK AT OUR WORK
THEY COULD TELL EXACTLY HOW WE VERIFIED THE IDENTITY.
OKAY, ON THIS SECOND IDENTITY WE MAY THINK TO MULTIPLY THIS
BY THE CONJUGATE,
BY MULTIPLYING THE NUMERATOR AND DENOMINATOR
BY 1 - SINE THETA.
HOWEVER, A LOT OF TIMES
WHEN YOU HAVE A TRIG FUNCTION SQUARED,
WE WANT TO USE ONE OF THOSE PYTHAGOREAN IDENTITIES.
AND WE KNOW THAT SINE SQUARED THETA
+ COSINE SQUARED THETA = 1,
SO IF WE SOLVE THAT IDENTITY FOR COSINE SQUARED THETA,
WE COULD REPLACE THIS WITH 1 - SINE SQUARED THETA.
AND ALREADY THE LEFT SIDE IS LOOKING
MORE LIKE THE RIGHT SIDE,
AND NOW OUR NUMERATOR
IS ACTUALLY A DIFFERENCE OF SQUARES,
AND THIS CAN BE FACTORED.
AND THE TWO FACTORS WOULD BE 1 + SINE THETA
AND 1 - SINE THETA.
AND AGAIN, WE CAN SEE THIS FRACTION SIMPLIFIES.
WE HAVE A COMMON FACTOR OF 1 + SINE THETA,
AND WE HAVE NOW VERIFIED OUR IDENTITY.
WE'LL WRITE IT ONE MORE TIME.
1 - SINE THETA = 1 - SINE THETA.
LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE MORE.
NOW AGAIN, WE'RE GOING TO WORK
WITH THE MORE COMPLICATED SIDE,
BUT THE QUESTION REALLY BECOMES
WHAT CAN WE DO TO THE LEFT SIDE
TO MAKE IT LOOK LIKE THE RIGHT SIDE?
OFTEN THERE ARE SEVERAL WAYS TO VERIFY AN IDENTITY.
WHAT COMES TO MIND FOR ME HERE
IS MULTIPLYING THIS FRACTION BY THE CONJUGATE.
LET'S GO AHEAD AND DO THAT FIRST,
AND FROM HERE WE'LL REWRITE THE REST.
SO WHEN WE FIND THIS PRODUCT,
WE'LL LEAVE THE NUMERATOR IN FACTORED FORM.
SO WE'LL HAVE SINE THETA x 1 - COSINE THETA.
AND THEN WHEN WE MULTIPLY THE DENOMINATOR OUT,
AND THEN WHEN WE FOIL THIS,
WE'RE GOING TO HAVE 1 - COSINE SQUARED THETA.
AND THEN AGAIN,
IT'S IMPORTANT TO CONTINUE TO REWRITE THIS EVERY TIME.
NOW WE HAVE 1 - COSINE SQUARED THETA IN THE DENOMINATOR,
AND WE CAN REPLACE THIS WITH SINE SQUARED THETA,
SO LET'S GO AHEAD AND DO THAT.
NEXT, THIS FRACTION SIMPLIFIES NICELY.
THERE'S A COMMON FACTOR OF SINE IN THE NUMERATOR
AND DENOMINATOR.
BUT NOTICE NOW WE HAVE A COMMON DENOMINATOR
OF SINE THETA,
SO WE CAN NOW COMBINE, OR ADD, THESE TWO FRACTIONS TOGETHER.
SO IT LOOKS LIKE WE HAVE 1 + 1.
THAT WOULD GIVE US 2.
AND THEN WE HAVE -COSINE THETA + COSINE THETA.
THAT WOULD = 0.
SO WE HAVE THE 2 IN THE NUMERATOR,
SINE THETA IN THE DENOMINATOR,
AND IN FACT 2 AND 2 / SINE THETA = 2 COSECANT THETA.
AND WE HAVE OUR VERIFICATION.
REMEMBER THAT 1/SINE THETA = COSECANT THETA.
OKAY. LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE MORE.
NOW, WE MAY QUESTION WHICH SIDE TO WORK WITH HERE,
SEEING AS SINE SQUARED THETA HERE
WOULD LEAD US TO PERFORM A SUBSTITUTION
USING THE PYTHAGOREAN IDENTITY.
SO WE CAN REPLACE THIS
WITH 1 - COSINE SQUARED THETA / COSINE THETA
= SECANT THETA - COSINE THETA.
NOW IN THIS FRACTION, BECAUSE WE'RE DIVIDING BY ONE TERM,
WE CAN BREAK THIS UP INTO TWO DIFFERENT FRACTIONS.
SO THIS IS THE SAME
AS 1/COSINE THETA - COSINE SQUARED THETA / COSINE THETA.
NOW THINGS ARE LOOKING GOOD HERE, I KNOW.
1/COSINE THETA = SECANT THETA.
THIS FRACTION SIMPLIFIES WITH A COMMON FACTOR
OF COSINE THETA.
THIS SIMPLIFIES TO 1.
WE'D HAVE ONE FACTOR OF COSINE THETA HERE,
AND THAT LOOKS LIKE ALL THAT WE NEED TO HAVE VERIFICATION.
LET'S GO AHEAD AND TRY ONE MORE.
AND AGAIN, YOU MAY THINK HERE TO DO A SUBSTITUTION
WITH SINE SQUARED THETA,
BUT THE OTHER OPTION WOULD BE TO TRY TO FACTOR
THE LEFT SIDE.
AND BECAUSE FACTORING THIS WOULD BE 2 SINE SQUARED
AND COSINE SQUARED,
I THINK THIS MAY BE THE WAY TO GO.
LET'S TRY IT. SO THAT'S A DIFFERENCE OF SQUARES.
WE HAVE SINE SQUARED THETA IN THE FIRST POSITION,
AND THEN COSINE SQUARED IN THE SECOND POSITION.
REMEMBER WHEN WE HAVE A DIFFERENCE OF SQUARES,
ONE IS PLUS AND ONE IS MINUS.
NEXT, SINE SQUARED THE DIFFERENCE
COSINE SQUARED THETA = 1.
SO ON THE LEFT SIDE WE NOW HAVE
JUST SINE SQUARED THETA - COSINE SQUARED THETA
= 2 SINE SQUARED THETA - 1.
NOTICE THE RIGHT SIDE IS IN TERMS OF SINE SQUAREDS
AND CONSTANTS.
WELL, HERE WE HAVE A SINE SQUARED.
WE CAN DO ANOTHER PYTHAGOREAN SUBSTITUTION
USING COSINE SQUARED THETA.
REMEMBER THAT COSINE SQUARED THETA
= 1 - SINE SQUARED THETA.
SINCE WE'RE SUBTRACTING COSINE SQUARED THETA,
WE NEED TO SUBTRACT
THE QUANTITY 1 - SINE SQUARED THETA.
LET'S GO AHEAD AND CLEAR THESE PARENTHESES NOW.
WE HAVE SINE SQUARED THETA - -SINE SQUARED THETA.
THAT'S 2 SINE SQUARED THETA.
- 1, AND WE HAVE VERIFICATION.
OKAY, SO WHEN YOU'RE WORKING ON THESE PROBLEMS,
IF YOU STRUGGLE, IT'S ONLY NATURAL.
IF YOU TRY SOMETHING AND IT DOESN'T WORK,
THEN YOU JUST WANT TO TRY SOMETHING ELSE.
SO IF YOU THINK OF THIS AS A TRIGONOMETRIC PUZZLE
THAT YOU HAVE TO SOLVE,
HOPEFULLY YOU'LL ENJOY THE PROCESS,
AND WITH TIME YOU'LL GET BETTER AT IT.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.