Domain, Range, and Signs of Trigonometric Function
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Domain, Range, and Signs of Trigonometric Function


WELCOME TO A VIDEO THAT WILL
DISCUSS MORE PROPERTIES OF TRIGONOMETRIC FUNCTIONS. THE GOALS OF THE VIDEO
ARE TO DETERMINE THE DOMAIN AND RANGE
OF THE TRIG FUNCTIONS AND ALSO TO DETERMINE THE SINE
OF THE TRIGONOMETRIC FUNCTIONS IN EACH QUADRANT. REMEMBER YOU CAN THINK
OF THE DOMAIN OF A FUNCTION AS THE INPUTS AND THE RANGE
AS THE FUNCTIONS OUTPUT. FOR TRIGONOMETRIC FUNCTIONS,
THE INPUTS ARE ANGLE MEASURES AND THE OUTPUTS ARE RATIOS OF
THE SIDES OF RIGHT TRIANGLES. SO LET’S BEGIN BY CONSIDERING
THE SINE AND COSINE FUNCTIONS. TO ME THE EASIEST WAY WOULD BE
TO ANALYZE THE GRAPH NOT ON THE UNIT CIRCLE BUT ON
THE X AND Y COORDINATE PLANE. SO LET’S GO AHEAD
AND TAKE A LOOK AT THAT. LET’S RIGHT NOW FOCUS ONLY ON
THESE FIRST TWO GRAPHS OF SINE AND COSINE, ESSENTIALLY WHAT YOU’RE
GOING TO SEE HERE AS — POINT ANIMATES AROUND
THE UNIT CIRCLE IT WILL PLOT THE FUNCTION
VALUES ON THE X AND Y AXIS AS WE SEE HERE. NOW THIS IS IN RADIANS BUT IN DEGREES IT WOULD BE
FROM 0 TO 360 DEGREES. LET’S SEE WHAT WE FIND. AGAIN JUST LOOK AT COSINE
AND SINE FOR RIGHT NOW. SO IF WE LOOK AT THESE
TWO GRAPHS, YOU CAN SEE
THAT IF WE CONTINUED AROUND THE CIRCLE AGAIN, THE GRAPH WOULD JUST CONTINUE OR IF WE WENT CLOCKWISE
OR FOR NEGATIVE ANGLES, WE WOULD GO BACKWARDS. SO THIS GRAPH WILL GO LEFT
AND RIGHT FOREVER. SO THE DOMAIN WOULD BE
ALL REELS. HOWEVER WE CAN SEE THAT
THE RANGE, THE HIGH POINTS AND THE LOW POINTS, THE LOWEST POINT IS -1
AND THE HIGHEST POINT IS +1. SO THAT WOULD BE OUR RANGE
FROM -1 TO +1. SO THE DOMAIN FOR BOTH SINE
AND COSINE WOULD BE ALL REELS BUT THE RANGE WE HAVE
A MINIMUM OF -1 AND A MAXIMUM OF +1. NOW LET’S TAKE A LOOK
AT COSECANT AND SECANT. NOW HERE IT’S GOING TO BE
A LITTLE MORE INVOLVED BECAUSE IF YOU REMEMBER THAT COSECANT THETA
IS EQUAL TO R/Y. COSECANT IS NOT GOING TO BE
DEFINED WHEN Y IS EQUAL TO 0. AND WE LOOK AT SECANT THETA. SECANT IS NOT GOING TO BE
DEFINED WHEN X IS EQUAL TO 0. SO LET’S TAKE A LOOK
AT COSECANT FIRST. WHEN WOULD Y=0? WELL Y WOULD EQUAL 0 HERE
AND HERE ON THE UNIT CIRCLE. SO WHAT ANGLES
WOULD THOSE REPRESENT? WELL THERE WOULD BE
ANY MULTIPLE OF 180 DEGREES. SO WHAT WE CAN SAY
FOR THE DOMAIN IS THAT THETA SUCH THAT THETA
CANNOT EQUAL 180 N OR SUM MULTIPLE OF 180 OR N
IS SUM INTEGER Z. AND WE’LL COME BACK
TO THE RANGE. LET’S TAKE A LOOK
AT THE DOMAIN FOR SECANT. THE SAME ISSUE NOW,
EXCEPT X CAN’T BE 0. AND X WOULD BE 0 HERE,
90 DEGREES IN AT 270 OR 90 PLUS OR MINUS
ANY MULTIPLE OF 180. SO WE CAN STATE THAT AS THETA
SUCH THAT THETA CANNOT EQUAL 90 LESS
ANY MULTIPLE OF 180 OR N WOULD BE SOME INTEGER. NOW LET’S GO BACK
TO THAT UNIT CIRCLE AND LOOK AT THE GRAPHS
OF COSECANT AND SECANT. HERE’S SECANT
AND HERE’S COSECANT. SO YOU CAN SEE
THERE ARE VERTICAL — AND FOR SECANT THEY OCCUR
AT 90 DEGREES PLUS ANY MULTIPLE OF 180 AND FOR COSECANT THEY OCCUR
AT 0, 180 AND SO ON. NOW THE RANGE, HOWEVER, YOU CAN SEE THERE’S
GOING TO BE TWO INTERVALS. IT’S GOING TO BE ONE OR LARGER
AND -1 OR BELOW. SO WE’RE GOING TO HAVE 1
TO INFINITY AND -1 TO -INFINITY
FOR THE RANGE. AND THAT’S TRUE FOR BOTH. SO -INFINITY TO -1
AND FROM 1 TO INFINITY AND THIS IS TRUE FOR BOTH. OKAY. WE’VE GOT TWO MORE
TO TALK ABOUT. LET’S TALK ABOUT TANGENT THETA
AND COTANGENT THETA. AND AGAIN WHAT WE’RE GOING
TO NOTICE THE TAN THETA IS NOT GOING
TO BE DEFINED WHEN X IS EQUAL TO 0 AND COTANGENT THETA IS NOT
DEFINED WHEN Y IS EQUAL TO 0. SO WE SHOULD BE ABLE TO FIGURE
OUT THE DOMAIN FOR TANGENT. AGAIN X WOULD BE 0
HERE AND HERE SO 90 PLUS
ANY MULTIPLE OF 180. SO THETA — THETA CANNOT EQUAL
90 + 180 N WHERE N IS SUM INTEGER. AND FOR COTANGENT Y WOULD BE 0
AT 0 180 AND ANY MULTIPLE. AND LET’S GO AHEAD
AND ANALYZE THE GRAPH AGAIN LOOKING AT TANGENT
AND COTANGENT. TANGENT AND COTANGENT
OCCUR HERE IN THE BOTTOM LEFT HAND CORNER AND AGAIN WE COULD SEE
THE VERTICAL ASYMPTOTES WHICH WE ALREADY IDENTIFIED
FOR TANGENT AND FOR COTANGENT WHEN Y IS EQUAL TO 0
ON THE UNIT CIRCLE. IF WE JUST LOOK AT ONE PIECE
OF THE GRAPH IT DOES GO UP FOREVER AND
DOWN FOREVER FOR BOTH GRAPHS SO OUR RANGE WOULD BE
ALL REELS.   OKAY, LET’S GO AHEAD
AND TAKE A LOOK AT DETERMINING THE SINE
OF THE TRIG FUNCTIONS IN EACH QUADRANT. NOW ON THE LEFT HERE YOU SEE
THE DEFINITIONS OF THE TRIG FUNCTIONS
IN TERMS OF X, Y’S AND R’S. THIS CAN BE VERY HELPFUL IN
DETERMINING WHICH ARE POSITIVE IN WHICH QUADRANT BECAUSE WE KNOW THAT
IN THE FIRST QUADRANT X AND Y ARE BOTH POSITIVE. IN THE SECOND, X IS NEGATIVE,
Y IS POSITIVE AND SO ON. AND WE KNOW THAT R
IS ALWAYS POSITIVE. SO THE FIRST THING,
THE FIRST QUADRANT IS EASIEST. IF X AND Y ARE POSITIVE,
THEY WILL ALL BE POSITIVE. IN THE SECOND QUADRANT, ANYTHING INVOLVING X
WOULD BE NEGATIVE. SO LOOKING AT THE LEFT HERE,
COSINE THETA, SECANT THETA, TANGENT THETA, AND COTANGENT
THETA ALL INVOLVE AN X. THEREFORE THEY WILL
ALL BE NEGATIVE IN THE SECOND QUADRANT SINE
AND COSECANT WILL BE POSITIVE. IN THE THIRD QUADRANT, BOTH THE X AND THE Y
COORDINATES ARE NEGATIVE SO ANYTHING INVOLVING
AN X OR A Y WOULD BE NEGATIVE UNLESS IT INVOLVED
BOTH AN X AND A Y. SO IF YOU TAKE A LOOK
AT TANGENT THETA AND COTANGENT THETA, WE’D HAVE A NEGATIVE
DIVIDED BY A NEGATIVE. THESE TWO WILL REMAIN
POSITIVE. ALL THESE OTHERS
ONLY INVOLVE AN X OR A Y. THEREFORE THESE WILL
ALL BE NEGATIVE. REMEMBER R IS ALWAYS POSITIVE. AND LASTLY
IN THE FOURTH QUADRANT, X IS POSITIVE
AND Y IS NEGATIVE. SO IF X IS POSITIVE SO IF COSINE AND SECANT
ARE POSITIVE, LET’S GO AHEAD
AND MARK THOSE OFF. SINCE Y IS NEGATIVE, THESE TWO WOULD BE
NEGATIVE SINE AND COSECANT AND LASTLY TANGENT AND
COTANGENT INVOLVE 1X AND 1Y SO THEY WILL BE NEGATIVE. SO THIS KIND OF EXPLAINS WHY
EACH OF THESE ARE POSITIVE, NEGATIVE IN EACH QUADRANT. THERE ARE SOME ACRONYMS
OUT THERE THAT CAN BE HELPFUL IN REMEMBERING THESE BUT I LIKE TO THINK OF IT
IN TERMS OF WHY THEY ARE BUT FOR EXAMPLE ASTC
HAS BEEN USED AND IT STANDS FOR ALL STUDENTS
TAKE CALCULUS. “A”” MEANING ALL OF THESE
TRIG FUNCTIONS ARE POSITIVE AND THE SECOND S MEANING
SINE AND IT’S RECIPROCAL ARE POSITIVE, THE REST ARE NEGATIVE. AND THE THIRD TANGENT AND
ITS RECIPROCAL ARE POSITIVE, THE REST ARE NEGATIVE
AND C FOR COSINE AND ITS RECIPROCAL
BEING POSITIVE AND THE REST NEGATIVE. YOU COULD ALSO THINK OF THIS
AS THE WORD ACTS, A-C-T-S. THIS WAS MORE
OF A DEFINITION VIDEO, BUT HOPEFULLY YOU FOUND IT
HELPFUL. THANK YOU AND HAVE A GOOD DAY.  

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