- WE WANT TO FIND THE EXACT TRIG 
FUNCTION VALUES
USING REFERENCE TRIANGLES.
NOTICE HOW THE ANGLES WERE GIVEN 
IN RADIANS,
SO WE COULD CONVERT THESE 
TO DEGREES.
BUT I THINK IT'S IMPORTANT 
TO LEARN
HOW TO WORK WITH RADIANS 
AS WELL.
SO LET'S SKETCH THESE ANGLES 
IN CENTER POSITION,
FORM THE REFERENCE TRIANGLE,
AND THEN DETERMINE THE TRIG 
FUNCTION VALUES.
SO WE'LL FIRST SKETCH 7 PI/4 
RADIANS IN STANDARD POSITION,
SO HERE'S THE INITIAL SIDE.
NOW, WE'LL START TO ROTATE 
COUNTER CLOCKWISE.
WELL, HALF A ROTATION COUNTER 
CLOCKWISE WOULD BE PI RADIANS.
BUT BECAUSE OUR ANGLE 
IS A MULTIPLE OF PI/4 RADIANS,
IT'S HELPFUL TO VIEW THIS 
AS 4 PI/4 RADIANS
WHICH MEANS FROM HERE,
WE NEED TO ROTATE ANOTHER 3 PI/4 
RADIANS.
WELL, 1 COMPLETE 
ROTATION COUNTER CLOCKWISE
WOULD BE 2 PI RADIANS WHICH IS 
EQUAL TO 8 PI/4 RADIANS.
SO WE COULD ALSO THINK OF THIS
AS WE HAVE TO ROTATE 
1 PI/4 RADIANS
LESS THAN 8 PI/4 RADIANS
WHICH MEANS WE WOULD ROTATE 
SOMEWHERE IN HERE
PI/4 RADIANS 
SHORT OF 1 COMPLETE ROTATION.
SO THIS WOULD BE THE TERMINAL 
SIDE.
AND WE SHOULD RECOGNIZE 
THAT THE ANGLE
BETWEEN THE TERMINAL SIDE AND 
THE X AXIS IS PI/4 RADIANS.
IF WE ROTATED ONE MORE PI/4 
RADIANS,
WE WOULD BE AT 8 PI/4 RADIANS.
SO NOW, WE CAN FORM 
OUR REFERENCE TRIANGLE
BY SKETCHING IT PERPENDICULAR 
TO THE X AXIS HERE.
SO HERE'S OUR REFERENCE TRIANGLE 
IN BLACK.
AND BECAUSE WE KNOW THAT PI/4 
RADIANS IS EQUAL TO 45 DEGREES,
WE KNOW WE LABEL THE 2 LEGS 
OF THIS RIGHT TRIANGLE AS 1
AND THE HYPOTENUSE AS SQUARE 
ROOT 2.
BUT WE ALSO HAVE TO MAKE SURE
THAT WE HAVE THE RIGHT SINES 
ON THE LEGS,
AND SINCE WE'RE IN THE 4th 
QUADRANT
WHERE THE X COORDINATE 
IS POSITIVE
AND THE Y COORDINATE 
IS NEGATIVE,
THIS 1 MUST BE NEGATIVE.
AND NOW, WE CAN USE 
THIS REFERENCE ANGLE
AND THIS REFERENCE TRIANGLE 
TO DETERMINE
THESE FIRST 2 TRIG FUNCTION 
VALUES.
SO THE SINE OF 7 PI/4 RADIANS 
IS EQUAL TO THE RATIO
OF THE OPPOSITE SIDE 
TO THE HYPOTENUSE
WHICH IS -1 DIVIDED BY SQUARE 
ROOT 2.
SO THIS IS THE EXACT VALUE 
FOR THIS TRIG FUNCTION.
BUT SOMETIMES WE ARE ASKED TO 
RATIONALIZE THE DENOMINATOR,
SO LET'S ALSO DO THAT.
TO RATIONALIZE THE DENOMINATOR,
WE MULTIPLY BY THE SQUARE ROOT 
OF 2/THE SQUARE ROOT OF 2,
SO THIS WOULD BE -SQUARE ROOT 2 
DIVIDED BY 2.
SO AGAIN 
DEPENDING ON YOUR DIRECTIONS,
YOU MAY BE ABLE TO EXPRESS
YOUR ANSWER IN THIS FORM 
OR THIS FORM.
AND NOW FOR THE COSINE 
OF 7 PI/4 RADIANS,
WE NEED THE RATIO OF THE 
ADJACENT SIDE TO THE HYPOTENUSE
WHICH WOULD BE 1 
DIVIDED BY SQUARE ROOT 2.
SO IF WE RATIONALIZE THE 
DENOMINATOR LIKE WE DID UP HERE,
THE DIFFERENCE IS WE WOULD HAVE 
SQUARE ROOT 2/2.
SO AGAIN THESE ALWAYS ARE EQUAL,
ONE HAS AN IRRATIONAL 
DENOMINATOR
AND ONE HAS A RATIONAL 
DENOMINATOR.
NOW FOR THE NEXT 2,
WE'LL SKETCH -3 PI/4 RADIANS 
AND FORM A REFERENCE TRIANGLE.
SO BECAUSE THEIR ANGLE 
IS NEGATIVE,
WE'LL NOW ROTATE CLOCKWISE 
3 PI/4 RADIANS.
WELL, ONE-QUARTER ROTATION 
CLOCKWISE
WOULD BE -PI/2 RADIANS.
BUT BECAUSE OUR ANGLE IS A 
MULTIPLE OF PI/4 RADIANS,
IT'S HELPFUL TO VIEW THIS 
AS -2 PI/4 RADIANS
WHICH MEANS WE NEED TO ROTATE 
ANOTHER PI/4 RADIANS CLOCKWISE
WHICH WOULD BRING US TO HERE,
SO HERE'S OUR TERMINAL SIDE.
SO IF THIS IS -3 PI/4 RADIANS 
AND HALF A ROTATION
WOULD BE -PI OR -4 PI/4 RADIANS,
WE KNOW THAT OUR REFERENCE ANGLE 
HERE MUST BE PI/4 RADIANS.
SO NOW, WE'LL FORM THE REFERENCE 
TRIANGLE.
AGAIN BECAUSE WE HAVE A PI/4,
PI/4 RIGHT TRIANGLE OR 45-45 
RIGHT TRIANGLE,
WE CAN LABEL THE 2 LEGS 1, 
THE HYPOTENUSE SQUARE ROOT 2.
BUT NOW, WE'RE IN THE 3rd 
QUADRANT
WHERE BOTH THE X COORDINATE AND 
THE Y COORDINATE ARE NEGATIVE,
SO THIS WILL BE -1 
AND SO WILL THIS.
SO THE SINE OF -3 PI/4 RADIANS
IS EQUAL TO THE RATIO OF THE 
OPPOSITE SIDE TO THE HYPOTENUSE,
SO WE'D HAVE -1 DIVIDED 
BY SQUARE ROOT 2,
OR IF WE WANT TO RATIONALIZE 
THE DENOMINATOR,
WE WOULD HAVE -SQUARE ROOT 2/2.
AND NOTICE FOR THE COSINE 
FUNCTION VALUE,
WE'D HAVE THE RATIO 
OF THE ADJACENT SIDE
OF THE HYPOTENUSE WHICH IS STILL 
-1 DIVIDED BY SQUARE ROOT 2,
OR -SQUARE ROOT 2/2.
THAT'S GOING TO DO IT FOR THESE 
TWO EXAMPLES.
BUT I WILL LEAVE YOU 
WITH SOME NOTES ON THE 45-45-90
REFERENCE TRIANGLE OR THE PI/4, 
PI/4 REFERENCE TRIANGLE.
THIS IS THE TRIANGLE 
THAT WE USED,
BUT OF COURSE WE COULD MULTIPLY 
EACH LENGTH BY A CONSTANT.
SO IF THE CONSTANT WAS X,
WE COULD ALSO EXPRESS THIS 
REFERENCE TRIANGLE
IN THIS FORM HERE.
OKAY, I HOPE YOU 
FOUND THIS HELPFUL.
