- IN THIS VIDEO WE'LL BE EVALUATING
TRIGONOMETRIC EXPRESSIONS INVOLVING BOTH TRIG FUNCTIONS
AND INVERSE TRIG FUNCTIONS.
AND FOR THESE EXAMPLES
THE INNER FUNCTION WILL BE THE TRIG FUNCTION,
AND THE OUTER FUNCTION WILL BE THE INVERSE TRIG FUNCTION.
SO FOR THE FIRST EXAMPLE WE HAVE INVERSE SINE
OF SINE PI/3.
LET'S START BY MODELING THE SINE OF PI/3
WITH A REFERENCE TRIANGLE ON THE COORDINATE PLANE.
PI/3 IS EQUAL TO 60 DEGREES,
SO WE'LL SKETCH A 60 DEGREE REFERENCE ANGLE
AND REFERENCE TRIANGLE IN THE FIRST QUADRANT.
SO WE KNOW THE SHORT LEG IS 1.
THE HYPOTENUSE IS 2, AND THE LONGER LEG IS SQUARE ROOT 3.
AND THIS IS PI/3 RADIANS.
SO THE SINE OF PI/3 = SQUARE ROOT 3/2.
SO WE HAVE INVERSE SINE OF SQUARE ROOT 3/2.
AND NOW WE KNOW THAT THE INVERSE SINE FUNCTION
RETURNS AN ANGLE THETA,
WHERE THETA MUST BE ON THE CLOSED INTERVAL
FROM -PI/2 TO +PI/2.
WELL, THE ANGLE IN THIS INTERVAL
THAT HAS A SINE FUNCTION VALUE OF SQUARE ROOT 3/2
IS JUST PI/3,
THE ANGLE THAT WE JUST MODELED.
SO YOU MIGHT BE ASKING WHY GO THROUGH ALL OF THIS
WHEN WE KNOW THAT INVERSE SINE AND SINE
ARE INVERSES OF ONE ANOTHER
AND, THEREFORE, UNDO EACH OTHER?
WHILE THAT IS TRUE
SOMETIMES IT'S NOT QUITE AS STRAIGHTFORWARD
AS THIS FIRST EXAMPLE.
LET'S TAKE A LOOK AT OUR SECOND EXAMPLE.
WE HAVE INVERSE SINE OF COSINE -3PI/4.
LET'S GO AHEAD AND SKETCH THIS ANGLE -3PI/4.
WE'LL HAVE A CLOCKWISE ROTATION OF 135 DEGREES,
SO IT'S GOING TO LOOK SOMETHING LIKE THIS.
SO OUR REFERENCE TRIANGLE WOULD BE HERE.
SO IF THIS IS -135 OR -3PI/4,
THIS ANGLE HERE WOULD BE 45 DEGREES OR PI/4.
LET'S GO AHEAD AND LABEL IT AS 45 DEGREES.
SO THIS IS A 45, 45, 90 REFERENCE TRIANGLE.
SO WE'D HAVE -1, -1, AND SQUARE ROOT 2.
SO THE COSINE OF -3PI/4 IS GOING TO BE THE RATIO
OF THE ADJACENT SIDE OF THE HYPOTENUSE
OR -1/SQUARE ROOT 2.
SO THIS SIMPLIFIES TO INVERSE SINE OF -1/SQUARE ROOT 2.
SO NOW WE NEED AN ANGLE THAT HAS A SINE FUNCTION VALUE
OF -1 DIVIDED BY SQUARE ROOT 2,
BUT THIS ANGLE THETA DOES NEED TO BE ON THE CLOSE INTERVAL
FROM -PI/2 TO PI/2.
BECAUSE THERE ARE AN INFINITE NUMBER OF ANGLES
THAT HAVE A SINE FUNCTION VALUE OF -1
DIVIDED BY SQUARE ROOT 2,
SO THERE'S A RESTRICTION ON THE OUTPUT
OR THE RANGE OF INVERSE SINE.
SO THE ANGLE THAT'S MODELED HERE
IS NOT THE ANGLE THAT WE CAN USE
TO DETERMINE WHAT THETA SHOULD BE.
THETA HAS TO BE FROM ZERO TO PI/2 RADIANS
OR FROM ZERO TO -PI/2 RADIANS.
AND SINCE THE SINE FUNCTION VALUE IS NEGATIVE
WE'RE GOING TO HAVE TO SKETCH A 45, 45, 90 REFERENCE ANGLE
IN THE FOURTH QUADRANT NOT THE THIRD.
SO IT WOULD HAVE TO LOOK SOMETHING LIKE THIS.
NOTICE HOW THE Y COORDINATE WOULD STILL BE -1.
THE X COORDINATE'S NOT INVOLVED IN THE SINE FUNCTION
AND HYPOTENUSE IS SQUARE ROOT 2.
SO THIS ANGLE HERE IS THE ANGLE WE'RE LOOKING FOR
ON THIS INTERVAL,
SO IT'S GOING TO BE -45 DEGREES OR -PI/4 RADIANS.
AND THIS IS WHAT I MEAN
BY THEY'RE NOT ALL QUITE AS STRAIGHTFORWARD
AS THIS FIRST EXAMPLE.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE OF THESE.
HERE WE HAVE INVERSE TANGENT SINE -PI/6.
SO, AGAIN, WE'LL START WITH THE INNERMOST FUNCTION.
WE'RE GOING TO GO AHEAD AND SKETCH THE REFERENCE TRIANGLE
FOR SINE -PI/6.
PI/6 IS 30 DEGREES,
SO WE'RE GOING TO SKETCH -30 DEGREES IN STANDARD POSITION.
SOMETHING LIKE THIS.
SO HERE'S OUR REFERENCE TRIANGLE
TO HELP US EVALUATE SINE -PI/6.
SO WE'D HAVE -1 TO SQUARE ROOT 3.
SO THE SINE OF -PI/6 IS GOING TO BE THE RATIO
OF THE OPPOSITE SIDE TO THE HYPOTENUSE.
SO THIS IS GOING TO BE -1/2.
SO NOW WE HAVE INVERSE TANGENT OF -1/2.
AN INVERSE TANGENT RETURNS AN ANGLE THETA
ON THE OPEN INTERVAL FROM -PI/2 TO +PI/2.
AND, AGAIN, THIS IS NOT THE ANGLE WE'RE LOOKING FOR,
SO LET'S GO AHEAD AND CLEAR THIS.
BECAUSE THE TANGENT FUNCTION VALUE IS NEGATIVE
WE'LL HAVE TO SKETCH A REFERENCE TRIANGLE
IN THE FORTH QUADRANT
THAT HAS A TRIANGLE FUNCTION VALUE OF -1 DIVIDED BY 2.
SO IF THIS IS OUR ANGLE THETA, THE OPPOSITE SIDE WOULD BE -1.
THE ADJACENT SIDE WOULD BE +2,
AND WE DON'T KNOW THE HYPOTENUSE,
BUT WE DON'T REALLY NEED IT.
NOW, THIS IS NOT ONE OF THOSE NICE REFERENCE TRIANGLES,
BECAUSE THE 2 IS NOT THE HYPOTENUSE.
IT'S THE ADJACENT SIDE.
SO WE'RE ACTUALLY GOING TO HAVE TO USE OUR CALCULATOR
TO GET A DECIMAL APPROXIMATION OF THIS ANGLE,
KEEPING IN MIND THAT IT'S GOING TO BE A NEGATIVE ANGLE,
SO WE'RE LOOKING FOR AN ANGLE
THAT ROTATES FROM HERE TO HERE.
IT DOESN'T SAY WHETHER TO EXPRESS THIS IN DEGREES
OR RADIANS.
LET'S JUST MAKE SURE WE KNOW WHAT MODE WE'RE IN.
RIGHT NOW WE'RE IN DEGREE MODE.
LET'S GO AHEAD AND LEAVE IT IN DEGREE MODE FOR A MINUTE.
SO WE'RE GOING TO PRESS 2nd, TANGENT, -1/2
OR -1 DIVIDED BY 2.
SO OUR ANGLE IS APPROXIMATELY -26.6 DEGREES.
LET'S ALSO GET THE MEASURE IN RADIAN MODE.
SO WE'LL JUST GO AHEAD AND CHANGE THE MODE.
AND WE'LL TYPE IN 2nd,
TANGENT, -1 DIVIDED BY 2 AGAIN,
SO THAT WE HAVE BOTH MEASURES.
SO THETA CAN ALSO BE EXPRESSED IN RADIANS
AS APPROXIMATELY -0.46 RADIANS.
SO THIS WOULD BE APPROXIMATELY -26.6 DEGREES,
WHICH IS APPROXIMATELY EQUAL TO -0.46 RADIANS.
I HOPE YOU HAVE FOUND THESE EXAMPLES HELPFUL.