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- IN THESE EXAMPLES, WE'RE ASKED TO DETERMINE
THE QUADRANT OF THE TERMINAL SIDE OF THE ANGLE
BASED UPON INFORMATION WE KNOW ABOUT VARIOUS TRIG FUNCTIONS.
AND WE CAN ANSWER THESE QUESTIONS
BASED UPON INFORMATION FROM THE UNIT CIRCLE.
LET'S DO A QUICK REVIEW OF THE UNIT CIRCLE.
IF AN ANGLE IS SKETCHED IN THE CENTER POSITION,
THE POINT WHERE THE TERMINAL SIDE OF THE ANGLE
INTERSECTS THE UNIT CIRCLE
TELLS US INFORMATION ABOUT TRIG FUNCTION VALUES.
COSINE THETA WILL BE EQUAL TO THE X COORDINATE
AND THE SINE THETA WILL BE EQUAL TO THE Y COORDINATE.
IT'S ALSO TRUE THAT TANGENT THETA WOULD BE EQUAL TO Y/X
SINCE TANGENT THETA IS EQUAL TO SINE THETA/COSINE THETA.
AND JUST TO MAKE SURE WE REMEMBER,
WHEN BOTH THE X AND Y COORDINATES ARE POSITIVE
IN THIS REGION HERE, THIS IS QUADRANT 1.
WHEN X IS NEGATIVE AND Y IS POSITIVE, WE'RE IN QUADRANT 2.
AND IF BOTH X AND Y ARE NEGATIVE, WE'RE IN QUADRANT 3.
AND IF X IS POSITIVE AND Y IS NEGATIVE, WE'RE IN QUADRANT 4.
LET'S TAKE A LOOK AT OUR QUESTIONS.
HERE WE'RE GIVEN THAT COSINE THETA IS GREATER THAN 0.
WELL, THIS TELLS US THAT X HAS TO BE GREATER THAN 0
AND SINE THETA IS GREATER THAN 0,
SO Y WOULD ALSO BE POSITIVE.
SO IF IS POSITIVE AND Y IS POSITIVE,
WE WOULD BE IN QUADRANT 1.
ON OUR SECOND EXAMPLE,
WE'RE TOLD THAT COSINE THETA IS LESS THAN 0
WHICH MEANS X WOULD HAVE TO BE LESS THAN 0
AND SINE THETA IS GREATER THAN 0,
SO Y WOULD BE GREATER THAN 0.
SO X IS NEGATIVE AND Y IS POSITIVE.
SO GOING BACK TO OUR UNIT CIRCLE,
IF X IS NEGATIVE AND Y IS POSITIVE,
WE WOULD BE IN QUADRANT 2.
NOW, THESE NEXT TWO GET A LITTLE BIT MORE INVOLVED.
WE'RE GIVEN THAT COSINE THETA IS LESS THAN 0,
SO THAT TELLS US THAT X IS LESS THAN 0.
BUT NOW, WE'RE GIVEN THAT TANGENT THETA
IS GREATER THAN 0.
WELL, TANGENT THETA IS EQUAL TO Y DIVIDED BY X
AND WE'RE TOLD THIS IS GREATER THAN 0,
MEANING THAT IT'S POSITIVE.
SO IF X IS NEGATIVE AND Y DIVIDED BY X IS POSITIVE,
THAT MEANS Y WOULD ALSO HAVE TO BE NEGATIVE,
BECAUSE A NEGATIVE DIVIDED BY A NEGATIVE WOULD BE POSITIVE.
SO IF BOTH X AND Y ARE NEGATIVE,
WE WOULD BE IN QUADRANT 3.
FOR OUR LAST EXAMPLE,
WE'RE GIVEN THAT SECANT THETA IS GREATER THAN 0.
REMEMBER SECANT THETA IS RECIPROCAL OF COSINE THETA.
SO IF SECANT THETA IS POSITIVE,
COSINE THETA WOULD BE POSITIVE,
THEREFORE X HAS TO BE POSITIVE.
AND COSECANT THETA IS LESS THAN 0,
COSECANT THETA IS THE RECIPROCAL OF SINE THETA.
SO IF COSECANT THETA IS NEGATIVE,
THEN SINE THETA WOULD ALSO BE NEGATIVE,
THEREFORE Y HAS TO BE NEGATIVE.
SO HERE WE HAVE X IS POSITIVE AND Y IS NEGATIVE.
SO IF X IS POSITIVE, Y IS NEGATIVE,
WE'RE IN THE 4th QUADRANT.
I HOPE THESE EXAMPLES HELP.