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- IN THIS VIDEO WE'LL EVALUATE SEVERAL INVERSE TRIG EXPRESSIONS
EXPRESSING OUR ANSWER IN RADIANS.
SO FOR THE FIRST EXPRESSION WE HAVE INVERSE SINE
OF NEGATIVE SQUARE ROOT 3/2.
SO WE'RE LOOKING FOR AN ANGLE
THAT SINE FUNCTION VALUE OF NEGATIVE SQUARE ROOT 3/2.
SO ONE OF THE MOST IMPORTANT THINGS
ABOUT EVALUATING THESE EXPRESSIONS
IS KNOWING THE INTERVAL FOR THE OUTPUT OR THE RANGE.
INVERSE SINE OR ARC SINE WILL ONLY RETURN AN ANGLE
IN THE CLOSED INTERVAL FROM -PI/2 TO PI/2.
SO THE FIRST THING WE NEED TO BE AWARE OF
IN ALL OF THESE EXPRESSIONS
IS WHAT IS THE INTERVAL FOR THE POSSIBLE OUTPUTS.
FOR INVERSE C-CANT, OR ARC C-CANT THE OUTPUT
IS AN ANGLE THETA IN THE SAME INTERVAL
AS INVERSE COSINE OR ARC COSINE.
SO WE HAVE THE CLOSED INTERVAL FROM ZERO TO PI
EXCEPT THETA CANNOT EQUAL PI/2
SINCE THE C-CANT FUNCTION IS UNDEFINED AT PI/2.
FOR ARC COSECANT THE OUTPUT OR RANGE FOR ARC COSECANT
WOULD BE THE SAME AS THE OUTPUT FOR INVERSE SINE
OR ARC SINE MEANING ON THE CLOSED INTERVAL
FROM -PI/2 TO PI/2,
EXCEPT THETA CAN'T EQUAL ZERO RADIANS
BECAUSE THE C-CANT FUNCTION IS UNDEFINED AT ZERO RADIANS.
AND FOR INVERSE COTANGENT OR ARC COTANGENT
THIS EXPRESSION WOULD RETURN AN ANGLE THETA
IN THE OPEN INTERVAL FROM ZERO TO PI RADIANS.
NOW, SOME TEXTBOOKS DO USE THE INTERVAL FROM -PI/2 TO PI/2,
BUT THIS INTERVAL, I THINK, IS A LITTLE BIT MORE COMMON.
NOW, LET'S GO AHEAD AND COMPLETE OUR REFERENCE TRIANGLE.
HERE WE HAVE THE 30/60/90 TRIANGLE.
WE COULD LABEL THE SHORT LEG ONE,
THE HYPOTENUSE TWO
AND THE LONGER LEG SQUARE ROOT THREE.
AND FOR THE 45/45/90, WE WOULD HAVE 1/1/SQUARE ROOT TWO.
SO LET'S GO AHEAD AND EVALUATE THESE.
FOR INVERSE SINE NEGATIVE SQUARE ROOT 3/2
WE'RE LOOKING FOR AN ANGLE THAT HAS A SINE FUNCTION VALUE
OF NEGATIVE SQUARE ROOT 3/2 IN THIS INTERVAL.
SO LET'S TAKE A LOOK AT OUR REFERENCE TRIANGLES FIRST.
NOTICE THAT THE SINE OF 60 DEGREES IS SQUARE ROOT 3/2.
WHICH MEANS 60 DEGREES IS OUR REFERENCE TRIANGLE.
BUT SINCE WE NEED A NEGATIVE SINE FUNCTION VALUE
WE'RE GOING TO HAVE A 60 DEGREE REFERENCE ANGLE
IN THE FORTH QUADRANT.
SO OUR REFERENCE TRIANGLE WOULD LOOK SOMETHING LIKE THIS
WHERE THE Y COORDINATE WOULD BE NEGATIVE SQUARE ROOT THREE.
SO THE SINE OF THIS ANGLE HERE,
OUR REFERENCE ANGLE IS NEGATIVE SQUARE ROOT OF 3/2.
THEREFORE THIS WOULD BE A -60 ANGLE OR -PI/3 RADIANS.
FOR THE NEXT EXAMPLE WE'RE LOOKING FOR AN ANGLE
THAT HAS A C-CANT FUNCTION VALUE OF TWO.
WELL, IF WE WANT WE COULD WRITE THIS AS TWO OVER ONE.
AND THE REASON THAT'S HELPFUL
IS IF THE C-CANT FUNCTION VALUE IS 2/1
THEN INVERSE COSINE WOULD BE 1/2, THE RECIPROCAL.
SO IT WOULD BE A LITTLE BIT EASIER
FOR US TO RECOGNIZE THE ANGLE
IF WE USE INVERSE COSINE RATHER THAN INVERSE C-CANT.
AGAIN, GOING BACK TO OUR REFERENCE TRIANGLES,
A 60 ANGLE DOES HAVE A COSINE FUNCTION VALUE OF 1/2.
SO OUR REFERENCE ANGLE IS 60 DEGREES
AND SINCE THE COSINE FUNCTION VALUE IS POSITIVE
IT WILL BE IN THE FIRST QUADRANT.
THEREFORE OUR ANGLE IN THIS INTERVAL
IS GOING TO BE 60 DEGREES OR +PI/3 RADIANS.
INVERSE COSECANT 2 SQUARE ROOT 3 DIVIDED BY 3
MEANS YOU WANT TO FIND AN ANGLE
THAT HAS A COSECANT FUNCTION VALUE OF THIS RATIO
WHICH IS GOING TO BE A LITTLE BIT CHALLENGING.
BUT IF INVERSE COSECANT IS EQUAL TO THIS RATIO,
THEN INVERSE SINE WOULD BE EQUAL TO THE RECIPROCAL RATIO
OR 3 DIVIDED BY 2 SQUARE ROOT 3.
SO IF WE TAKE A MOMENT AND RATIONALIZE THIS
AND SIMPLIFY IT,
IT WILL HELP US RECOGNIZE OUR REFERENCE ANGLE.
SO LET'S GO AHEAD AND DO THAT UP HERE.
WE HAVE 3 DIVIDED BY 2 SQUARE ROOT 3
IF YOU WANT TO RATIONALIZE THIS.
SO WE'RE GOING TO HAVE 3 SQUARE ROOT 3 ALL OVER.
THIS WILL BE 2 x 3 WHICH IS 6.
SO THIS SIMPLIFIES TO SQUARE ROOT 3/2.
SO THIS IS INVERSE SINE OF SQUARE ROOT 3/2.
AGAIN, GOING BACK TO OUR REFERENCE TRIANGLES,
NOTICE THAT 60 DEGREES HAS A SINE FUNCTION VALUE
OF POSITIVE SQUARE ROOT 3/2.
SIXTY DEGREES IS A REFERENCE ANGLE AND WE ALSO WANT --
AND IT'S ALSO IN THE FIRST QUADRANT.
SO THE ANGLE WE'RE LOOKING FOR IS 60 DEGREES OR +PI/3 RADIANS.
AND FOR THE LAST EXAMPLE WE HAVE INVERSE COTANGENT -1.
SO WE WANT AN ANGLE THAT HAS A COTANGENT FUNCTION VALUE OF -1.
WELL, A 45 DEGREE ANGLE
DOES HAVE A COTANGENT FUNCTION VALUE OF +1.
SO 45 DEGREES WILL BE OUR REFERENCE ANGLE.
BUT SINCE WE WANT AN ANGLE IN THIS INTERVAL
IT'S GOING TO HAVE TO BE IN THE SECOND QUADRANT.
SO WE'LL SKETCH A 45 DEGREE REFERENCE ANGLE
IN THE SECOND QUADRANT.
IT WILL BE HERE.
NOTICE HOW THE X COORDINATE WOULD BE NEGATIVE.
THE Y COORDINATE WOULD BE POSITIVE,
AND THIS WOULD BE SQUARE ROOT TWO.
SO THE ANGLE WE'RE LOOKING FOR ON THIS INTERVAL
WOULD BE FROM HERE TO HERE OR 135 DEGREES.
AND 135 DEGREES IS THE SAME AS 3PI/4 RADIANS.
AND JUST BE AWARE ON THIS LAST ONE,
IF YOUR TEXTBOOK DEFINES THE OUTPUT OR THE RANGE
FOR INVERSE COTANGENT AS THETA ON THE INTERVAL
FROM -PI/2 TO +PI/2,
THEN INSTEAD OF 3PI/4,
WE'D BE OVER HERE IN THE FORTH QUADRANT AT -PI/4.
SO BE SURE AND CHECK YOUR TEXTBOOK ON THEIR DEFINITION
OF THE OUTPUT OR RANGE FOR INVERSE COTANGENT.
I HOPE YOU FOUND THIS HELPFUL.