- IN THESE NEXT TWO EXAMPLES
WE'RE ASKED TO REWRITE A TRIG EXPRESSION
USING A HALF ANGLE IDENTITY.
HERE WE HAVE SINE OF 5X RAISED TO THE 2nd POWER
SO WE'LL BE APPLING THE HALF ANGLE IDENTITY FOR SINE.
SO THERE'S A COUPLE OF THINGS WE HAVE TO REMEMBER HERE.
FIRST, NOTICE HOW THIS FUNCTION IS BEING SQUARED
SO IT'S NOT A PERFECT FIT.
IF WE WANT TO REWRITE A SQUARE ROOT USING A RATIONAL EXPONENT,
REMEMBER THE SQUARE ROOT OF X IS EQUAL TO X TO THE POWER OF 1/2.
SO LETS START BY TAKING THIS HALF ANGLE IDENTITY
AND SQUARING BOTH SIDES
AND WE'LL ALSO REWRITE THIS SQUARE ROOT
USING A RATIONAL EXPONENT
SO WE'LL HAVE + OR - THIS WILL BE 1 - COSINE "A" DIVIDED BY 2
RAISED TO THE 1/2 POWER SQUARED.
THE NEXT RULE WE HAVE TO REMEMBER
IS WHEN WE HAVE A POWER TO A POWER
WE MULTIPLY OUR EXPONENTS.
SO 1/2 x 2 IS GOING TO EQUAL 1 AND SINCE WE'RE SQUARING THIS,
THIS + OR - GOES AWAY
BECAUSE IF WE SQUARE A POSITIVE IT'S POSITIVE
AND IF WE SQUARE A NEGATIVE IT'S STILL POSITIVE.
SO WE HAVE THE SINE OF A/2 SQUARED
IT'S GOING TO BE EQUAL TO 1 - COSINE "A" DIVIDED BY 2.
SO THE NEXT STEP IS TO DETERMINE
WHAT IS THE VALUE OF "A" GIVEN SINE OF 5X SQUARED.
WELL 5X HAS TO EQUAL A/2.
SO IF WE SET A/2 = 5X
AND MULTIPLY BOTH SIDES BY 2 WE KNOW THAT "A" IS EQUAL TO 10X.
SO USING THE HALF ANGLE IDENTITIES
"A" WILL ALWAYS BE TWICE THE GIVEN ANGLE.
SO NOW WE HAVE ALL THE INFORMATION
WE NEED TO APPLY THIS SLIGHT VARIATION
OF THE HALF ANGLE IDENTITY FOR SINE.
WE'RE GOING TO HAVE SINE OF 5X RAISED TO THE 2nd
IT'S GOING TO BE EQUAL TO 1 - COSINE 10X DIVIDED BY 2
AND IF WE WANT WE CAN REWRITE THIS AS 1/2 - 1/2 COSINE 10X.
SO THERE'S REALLY NOT A LOT GOING ON HERE
OTHER THAN GETTING FAMILIAR
WITH THESE NEW HALF ANGLE IDENTITIES.
LET'S TAKE A LOOK AT ONE MORE.
HERE WE HAVE COSINE OF 2X RAISED TO THE 4th POWER
SO LET'S START BY TAKING THIS HALF ANGLE IDENTITY FOR COSINE
AND RAISING IT TO THE 4th POWER.
SO WE'LL HAVE COSINE OF A/2
AND WE'RE GOING TO RAISE IT TO THE 4th POWER THIS TIME.
IT'S GOING TO BE EQUAL TO + OR -
AND WE'LL ALSO WRITE THIS AS A RATIONAL EXPONENT AGAIN
SO WE'LL HAVE + OR - THIS WILL BE 1 + COSINE "A" DIVIDED BY 2
TO THE 1/2 POWER AND THEN AGAIN WE'RE RAISING BOTH SIDES
TO THE 4th POWER.
WELL HERE WE'RE GOING TO HAVE 1/2 x 4 THAT'S GOING TO BE 2.
SO WE HAVE COSINE OF A/2 RAISED TO THE 4th
IT'S GOING TO BE EQUAL TO 1 + COSINE "A" ALL OVER 2
RAISED TO THE 2nd POWER.
AGAIN, NOTICE HOW THE + OR - SYMBOL IS NO LONGER NEEDED
BECAUSE WE'RE RAISING IT TO AN EVEN POWER
AND REMEMBER TO SQUARE A FRACTION
WE CAN SQUARE THE DENOMINATOR AND SQUARE THE NUMERATOR.
SO LET'S GO AHEAD AND MULTIPLY THIS OUT.
OUR DENOMINATOR IS GOING TO BE 4.
IF WE MULTIPLY THIS OUT,
WE'RE GOING TO HAVE 1 + COSINE "A" x 1 + COSINE "A"
SO WE'LL HAVE 1 + COSINE "A" + COSINE "A."
THAT'S 2 COSINE "A"
AND THEN WE'LL HAVE COSINE SQUARED "A."
NOW WE'LL GO AHEAD
AND APPLY THIS SLIGHT VARIATION OF THE HALF ANGLE IDENTITY
TO THE GIVEN EXPRESSION
AND REMEMBER IF A/2 IS EQUAL TO 2X
THE ANGLE AND THE HALF ANGLE IDENTITY
IF WE MULTIPLY BOTH SIDES BY 2 WE'LL HAVE "A" = 4X.
SO WHEN WE APPLY THIS IDENTITY
WE'LL HAVE COSINE OF 2X RAISED TO THE 4th POWER.
IT'S GOING TO BE EQUAL TO 1 + 2 COSINE 4X + COSINE SQUARED 4X
ALL DIVIDED BY 4
AND WE COULD FACTOR OUT THE 1/4 TO HAVE 1/4 x THE QUANTITY
1 + 2 COSINE 4X + COSINE SQUARED 4X
AND OF COURSE, WE COULD ALSO DISTRIBUTE THE 1/4
BUT I THINK WE'LL GO AHEAD AND LEAVE IT LIKE THIS.
AGAIN, OTHER THAN DOING SOME MANIPULATIONS
TO THESE HALF ANGLE IDENTITIES
THERE'S NOT A LOT GOING ON HERE
OTHER THAN BECOMING MORE FAMILIAR WITH THESE IDENTITIES.