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- IN THIS VIDEO WE'LL EVALUATE INVERSE TRIG EXPRESSIONS
INVOLVING INVERSE COSECANT, INVERSE SECANT
AND INVERSE COTANGENT
USING THE UNIT CIRCLE.
SO TO EVALUATE INVERSE COSECANT OF NEGATIVE SQUARE ROOT TWO
WE'RE LOOKING FOR AN ANGLE
THAT HAS A COSECANT VALUE OF NEGATIVE SQUARE ROOT TWO.
BUT IT'S IMPORTANT THAT WE RECOGNIZE THAT THE OUTPUT
OR THE RANGE OF THE INVERSE COSECANT
IS RESTRICTED TO AN ANGLE
IN THE CLOSE INTERVAL FROM -PI/2 TO PI/2
THE SAME AS INVERSE SINE
EXCEPT THETA CANNOT EQUAL ZERO.
BECAUSE THE COSECANT FUNCTION IS UNDEFINED
WHEN THETA IS EQUAL TO ZERO.
SO LET'S GO AHEAD AND LIST THE POSSIBLE OUTPUTS
FOR EACH OF THESE INVERSE EXPRESSIONS.
THE EXPRESSION INVERSE SECANT OR ARC SECANT OF -2
IS ONLY GOING TO RETURN AN ANGLE ON THE INTERVAL
THAT'S THE SAME AS INVERSE COSINE
WHICH MEANS AN ANGLE ON THE CLOSED INTERVAL
FROM ZERO TO PI RADIANS
BUT THETA CANNOT EQUAL PI/2
BECAUSE THE SECANT FUNCTION IS UNDEFINED AT PI/2.
AND THEN FOR INVERSE COTANGENT OR ARC COTANGENT,
THIS IS GOING TO RETURN AN ANGLE IN THE INTERVAL
SOMETIMES BASED UPON THE TEXTBOOK
THAT YOU'RE REFERRING TO.
MOST TEXTBOOKS REFER TO THE OUTPUT OR RANGE
OR THIS INVERSE TRIG FUNCTION
AS THE OPEN INTERVAL FROM ZERO TO PI RADIANS
BUT SOME TEXTBOOKS DO REFER TO THE TEXTBOOK AS THETA
ON THE OPEN INTERVAL FROM -PI/2 TO PI/2
WHERE THETA CAN'T EQUAL ZERO RADIANTS.
OKAY, LET'S GO AHEAD AND GO THROUGH ALL OF THESE
USING THE UNIT CIRCLE TO EVALUATE EACH EXPRESSION.
AND BECAUSE WE'RE GOING TO USE A UNIT CIRCLE
IT MIGHT BE HELPFUL TO REWRITE THESE INVERSE TRIG EXPRESSIONS
MEANING INVERSE COSECANT OF NEGATIVE SQUARE ROOT 2/1
IS THE SAME AS INVERSE SINE OF -1/SQUARE ROOT 2.
REMEMBER THE SINE FUNCTION AND THE COSECANT FUNCTION
ARE RECIPROCALS OF ONE ANOTHER.
AND SINCE WE'RE GOING TO USE THE UNIT CIRCLE
IT'S GOING TO BE EASIER TO LOCATION A Y COORDINATE
THAT WOULD BE EQUAL TO -1/SQUARE ROOT OF 2.
AND THEN IF WE RATIONALIZE THIS
WE WOULD HAVE NEGATIVE SQUARE ROOT 2/2.
SO WE'RE LOOKING FOR A Y COORDINATE
ON THE UNIT CIRCLE OF NEGATIVE SQUARE ROOT 2/2
WHERE THETA HAS TO BE IN THIS INVERSE.
SO ANGLE THETA MUST BE FROM ZERO TO PI/2 RADIANS
OR FROM ZERO TO -PI/2 RADIANS.
SO HERE'S THE Y COORDINATE THAT WE'RE LOOKING FOR
WHICH MEANS THE TERMINAL SIDE OF OUR ANGLE
MUST PASS THROUGH THIS POINT ON THE UNIT CIRCLE
WHERE THIS WOULD BE OUR INITIAL SIDE.
SO IN ORDER TO HAVE THIS TERMINAL SIDE IN THIS INTERVAL
WE'LL HAVE TO ROTATE CLOCKWISE 45 DEGREES.
THAT'S GOING TO BE -45 DEGREES OR -PI/4 RADIANS.
FOR THE SECOND EXAMPLE WE HAVE INVERSE SECANT OF -2
WHICH IS THE SAME AS -2/1.
SO EXCEPT THAT PI/2, INVERSE SECANT OF -2/1
WOULD BE EQUAL TO INVERSE COSINE OF -1/2, THE RECIPROCAL.
AND AGAIN, THIS IS HELPFUL IF WE'RE USING THE UNIT CIRCLE
BECAUSE NOW WE CAN LOOK FOR AN X COORDINATE
ON THE UNIT CIRCLE OF -1/2
ON THE INTERVAL FROM ZERO TO PI.
SO WE'RE LOOKING FOR AN X COORDINATE OF -1/2
ON THE INTERVAL FROM ZERO TO PI RADIANS.
AND SINCE THE X COORDINATE WOULD HAVE TO BE NEGATIVE,
WE ONLY HAVE TO LOOK IN THE SECOND QUADRANT.
AND HERE IS THE X COORDINATE OF -1/2.
SO THE TERMINAL SIDE OF OUR ANGLE
WOULD PASS THROUGH THIS POINT ON THE UNIT CIRCLE.
THERE IS THE INITIAL SIDE.
SO OUR ANGLE IS GOING TO BE 120 DEGREES OR 2PI/3 RADIANS.
AND FOR THE LAST EXAMPLE,
INVERSE COTANGENT SQUARE ROOT 3 OR SQUARE ROOT 3/1,
WE'RE LOOKING FOR AN ANGLE
THAT HAS A COTANGENT FUNCTION VALUE OF SQUARE ROOT 3/1.
WELL, ON THE UNIT CIRCLE
COTANGENT THETA IS EQUAL TO X DIVIDED BY Y.
SO WE'RE LOOKING FOR A POINT ON THE UNIT CIRCLE ON THIS INTERVAL
WHERE X DIVIDED BY Y WOULD BE EQUAL TO SQUARE ROOT THREE.
SO WE'RE LOOKING FOR AN ANGLE ON THIS INTERVAL
FROM ZERO TO PI RADIANS
WHERE X DIVIDED BY Y EQUALS POSITIVE SQUARE ROOT THREE.
SO WE ONLY HAVE TO LOOK IN THE FIRST QUADRANT.
LOOKING AT THIS POINT HERE
WE HAVE AN X COORDINATE OF SQUARE ROOT OF 3/2
AND A Y COORDINATE OF 1/2 OR 1 DIVIDED BY 2.
SO X DIVIDED BY Y WOULD BE SQUARE ROOT 3.
SO THE TERMINAL SIDE OF THE ANGLE
MUST PASS THROUGH THIS POINT ON THE UNIT CIRCLE.
SO OUR ANGLE THETA IS THREE DEGREES OR PI/6 RADIANS.
LET'S JUST GO AHEAD AND VERIFY THAT X DIVIDED BY Y
IS SQUARE ROOT 3.
WE WOULD HAVE SQUARE ROOT 3/2 DIVIDED BY 1/2
WHICH IS THE SAME AS SQUARE ROOT 3/2 x RECIPROCAL.
THE TWO SIMPLIFY OUT AND WE'RE LEFT WITH SQUARE ROOT 3.
I HOPE YOU FOUND THIS HELPFUL.