- IN THIS VIDEO WE'LL EVALUATE SEVERAL INVERSE TRIG EXPRESSIONS
USING THE UNIT CIRCLE.
WE COULD ALSO USE REFERENCE TRIANGLES
WHICH I WILL SHOW IN THE NEXT VIDEO.
THE FIRST EXPRESSION IS INVERSE SINE
OF NEGATIVE SQUARE ROOT 3/2.
AND THIS COULD ALSO BE EXPRESSED AS ARC SINE.
AND IF THIS IS WRITTEN AS ARC SINE,
IT MEANS EXACTLY THE SAME THING.
WE WANT TO FIND AN ANGLE THAT HAS A SINE FUNCTION VALUE
OF NEGATIVE SQUARE ROOT 3/2.
SO THE INPUT IS THE SINE FUNCTION VALUE,
AND THE OUTPUT IS GOING TO BE THE ANGLE.
ONE OF THE MOST IMPORTANT THINGS TO REMEMBER
ABOUT EVALUATING INVERSE TRIG EXPRESSIONS,
OR INVERSE TRIG FUNCTION
IS THAT THE OUTPUT OR THE RANGE HAS BEEN RESTRICTED.
THERE IS AN INFINITE NUMBER OF ANGLES
THAT HAVE A SINE FUNCTION VALUE OF NEGATIVE SQUARE ROOT 3/2,
HOWEVER, THE OUTPUT FOR INVERSE SINE, OR ARC SINE
IS RESTRICTED TO OR LIMITED TO THE CLOSED INVERSE
FROM -PI/2 TO +PI/2.
AND FOR INVERSE COSINE OR ARC SINE,
THE OUTPUT OR RANGE IS LIMITED TO THE CLOSED INTERVAL
FROM ZERO TO PI RADIANS.
AND FOR ARC TANGENT OR INVERSE TANGENT,
THE OUTPUT OR THE RANGE IS LIMITED TO THE OPEN INTERVAL
FROM -PI/2 TO PI/2.
AND THE REASON THIS INVERSE IS OPEN
BECAUSE REMEMBER THAT THE TANGENT FUNCTION IS UNDEFINED
AT -PI/2 AND PI/2.
THAT'S WHERE WE HAVE VERTICAL ASYMPTOTES.
SO TO EVALUATE INVERSE SINE OF NEGATIVE SQUARE ROOT 3/2
WE'RE GOING TO FIND THE ANGLE IN THIS INTERVAL
THAT HAS A SINE FUNCTION VALUE OF NEGATIVE SQUARE ROOT 3/2.
SO IF WE TAKE A LOOK AT THE UNIT CIRCLE
WE'RE ONLY GOING TO BE LOOKING AT THE FIRST QUADRANT
AND THE FOURTH QUADRANT ON THE INVERSE FROM -PI/2 TO PI/2.
AND ON THE UNIT CIRCLE SINE THETA = Y.
SO WE'RE LOOKING FOR A Y COORDINATE
OF NEGATIVE SQUARE ROOT 3/2 WHICH WE SEE HERE.
SO WE DO HAVE TO BE A LITTLE BIT CAREFUL HERE
BECAUSE THE ANSWER IS NOT 300 DEGREES OR 5PI/3 RADIANS
BECAUSE THE ANGLE HAS TO BE ON THIS INTERVAL.
BUT THIS IS THE TERMINAL SIDE OF THE ANGLE
THAT WE'RE LOOKING FOR.
SO HERE'S THE INITIAL SIDE AND HERE IS THE TERMINAL SIDE.
SO WE'D HAVE TO HAVE A CLOCKWISE ROTATION TO HERE.
SO INSTEAD OF RETURNING 300 DEGREES,
THE ANGLE IS GOING TO BE -60 DEGREES.
OR IF WE NEED TO EXPRESS OUR ANSWER IN RADIANS
THAT'S GOING TO BE -PI/3 RADIANS.
SO PROBABLY THE MOST IMPORTANT THING
ABOUT THESE TYPES OF PROBLEMS
IS KNOWING THE RESTRICTIONS ON THE OUTPUTS
OF THESE INVERSE TRIG EXPRESSIONS.
IF WE WEREN'T AWARE OF THE RANGE
OR THE OUTPUT OF THE INVERSE SINE FUNCTION
WE WOULDN'T KNOW WHICH ANGLE THIS EXPRESSION WOULD RETURN.
FOR INVERSE COSINE, AGAIN, WE'RE ONLY GOING TO BE LOOKING
ON THE UNIT CIRCLE FROM ZERO TO PI RADIANS
OR FROM HERE TO HERE.
AND BECAUSE WE'RE LOOKING FOR AN ANGLE
THAT HAS A COSINE FUNCTION OF ZERO,
AND ON THE UNIT CIRCLE OF COSINE THETA = X,
WE'RE NOW LOOKING FOR AN X COORDINATE OF ZERO.
AND HERE IT IS.
THEREFORE THE TERMINAL SIDE OF THE ANGLE
MUST INTERSECT THE UNIT CIRCLE AT THAT POINT.
SO HERE'S THE INITIAL SIDE.
HERE IS THE TERMINAL SIDE.
AND SO OUR ANGLE IS GOING TO BE 90 DEGREES OR PI/2 RADIANS.
NOW, FOR THE LAST EXAMPLE WE'RE LOOKING FOR AN ANGLE
THAT HAS A TANGENT FUNCTION VALUE OF -1
WHERE THE ANGLE MUST BE IN THIS INTERVAL.
SO WE'RE LOOKING FOR AN ANGLE FROM ZERO TO PI/2
OR FROM ZERO TO -PI/2.
AND ON THE UNIT CIRCLE
THE TANGENT FUNCTION WOULD BE Y DIVIDED BY X.
SO WE'RE LOOKING FOR A POINT WHERE Y DIVIDED BY X
WOULD BE EQUAL TO -1,
MEANING THE ABSOLUTE VALUE OF THE X AND Y COORDINATES
WOULD BE THE SAME,
BUT THEY ARE THE OPPOSITE SIGN.
SO HERE WE HAVE POSITIVE SQUARE ROOT 2/2,
AND POSITIVE SQUARE ROOT 2/2.
WHERE HERE THE TANGENT FUNCTION WOULD BE EQUAL TO POSITIVE ONE.
SO THAT'S NOT THE RIGHT POINT.
BUT DOWN HERE IN THE FORTH QUADRANT
WE HAVE AN X COORDINATE OF POSITIVE SQUARE ROOT 2/2,
AND A Y COORDINATE OR NEGATIVE SQUARE ROOT 2/2.
WELL, Y DIVIDED BY X WOULD BE - 1.
THEREFORE THIS IS THE POINT ON THE UNIT CIRCLE
THAT WE'RE LOOKING FOR
MEANING THE TERMINAL SIDE OF THE ANGLE WE'RE LOOKING FOR
PASSES THROUGH THIS POINT.
AGAIN, WE HAVE TO BE CAREFUL
BECAUSE WE'RE NOT LOOKING FOR 315 DEGREES OR 7PI/4 RADIANS
BECAUSE OF THE SPECIFIED OUTPUT FOR ARC TANGENT.
SO THE INITIAL SIDE WOULD BE HERE.
THE TERMINAL SIDE WOULD BE HERE.
BUT TO BE WITHIN THIS INTERVAL
WE'D HAVE TO HAVE A CLOCKWISE ROTATION.
AND IF THE COUNTER CLOCKWISE ROTATION IS 315 DEGREES,
THE CLOCKWISE ROTATION WOULD BE -45 DEGREES OR -PI/4 RADIANS.
I HOPE YOU HAVE FOUND THESE EXAMPLES HELPFUL.
AND AS I MENTIONED BEFORE
WE'LL TAKE A LOOK AT THESE SAME TYPES OF PROBLEMS
USING REFERENCE TRIANGLES INSTEAD IN THE NEXT VIDEO.