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- WE WANT TO DETERMINE THE EXACT TRIG FUNCTION VALUES
USING REFERENCE TRIANGLES.
SO WE'LL SKETCH EACH ANGLE IN STANDARD POSITION,
CONSTRUCT THE REFERENCE TRIANGLE,
THEN DETERMINE THE TRIG FUNCTION VALUE.
SO FIRST WE HAVE THE SINE OF 120 DEGREES.
SO WE'LL SKETCH 120 DEGREES IN STANDARD POSITION,
SO HERE'S THE INITIAL SIDE.
NOW, WE'LL ROTATE COUNTER CLOCKWISE 120 DEGREES.
WELL, IF WE ROTATE TO THE POSITIVE Y AXIS,
THIS WOULD BE 90 DEGREES.
SO WE NEED TO ROTATE 30 DEGREES MORE TO REACH 120 DEGREES.
SO RIGHT ABOUT HERE,
WOULD BE THE TERMINAL SIDE OF 120 DEGREES.
SO TO FORM THE REFERENCE TRIANGLE,
WE'LL SKETCH A PERPENDICULAR SEGMENT TO THE X AXIS HERE.
SO THIS BLACK TRIANGLE IS OUR REFERENCE TRIANGLE.
WELL, IF THIS IS 120 DEGREES
AND HALF A ROTATION COUNTER CLOCKWISE WOULD BE 180 DEGREES
THAT MEANS OUR REFERENCE ANGLE
OR THE ANGLE FORMED BETWEEN THE TERMINAL SIDE OF OUR ANGLE
AND THE X AXIS WOULD BE 60 DEGREES.
THIS IS OUR REFERENCE ANGLE WHICH WE CAN USE TO DETERMINE
THE SINE AT 120 DEGREES.
AND SINCE OUR REFERENCE TRIANGLE IS A 30-60-90 RIGHT TRIANGLE,
WE CAN LABEL THE SHORT LEG 1, THE HYPOTENUSE 2,
AND THE LONGER LEG SQUARE ROOT 3.
BUT WE ALSO HAVE TO MAKE SURE WE HAVE THE CORRECT SINE
ON THE 2 LEGS BASED UPON THE QUADRANT THAT WE'RE IN.
BECAUSE WE'RE IN QUADRANT 2 WHERE THE X COORDINATE
IS NEGATIVE AND THE Y COORDINATE IS POSITIVE,
THE LEG WITH LENGTH 1 MUST BE -1.
SO THE SINE AT 120 DEGREES WOULD BE THE RATIO
OF THE OPPOSITE SIDE TO THE HYPOTENUSE
OR SQUARE ROOT 3 DIVIDED BY 2.
NEXT, WE HAVE THE COSINE OF 210 DEGREES,
SO WE'LL START BY SKETCHING 210 DEGREES IN STANDARD POSITION.
HERE'S OUR INITIAL SIDE.
NOW, WE ROTATE COUNTER CLOCKWISE 210 DEGREES.
WELL, HALF A ROTATION WOULD BE HERE
WHICH WOULD BE 180 DEGREES
WHICH MEANS NOW WE NEED TO ROTATE 30 DEGREES
MORE TO REACH 210 DEGREES.
SO THIS WOULD BE THE TERMINAL SIDE OF 210 DEGREES.
AND NOW, THE ANGLE FORMED BETWEEN THE X AXIS
AND OUR TERMINAL SIDE HERE IS 30 DEGREES,
SO THIS WILL BE OUR REFERENCE ANGLE.
AND NOW, WE'LL FORM THE REFERENCE TRIANGLE.
AGAIN, WE'LL SKETCH
A PERPENDICULAR SEGMENT TO THE X AXIS.
SO THIS BLACK RIGHT TRIANGLE IS OUR REFERENCE TRIANGLE.
AND AGAIN, NOTICE HOW WE HAVE A 30-60-90 RIGHT TRIANGLE.
SO ONCE AGAIN, WE CAN LABEL THE SHORT LEG 1, THE HYPOTENUSE 2,
AND THE LONGER LEG SQUARE ROOT 3. BUT AS ALWAYS,
WE NEED TO MAKE SURE THAT WE HAVE THE CORRECT SINE
ON THE 2 LEGS OF THE REFERENCE TRIANGLE.
NOW THAT WE'RE IN THE 3rd QUADRANT,
WHERE BOTH THE X COORDINATE AND THE Y COORDINATE ARE NEGATIVE,
BOTH LEGS ARE NEGATIVE,
SO THE COSINE OF 210 DEGREES USING OUR REFERENCE TRIANGLE
WILL BE THE RATIO OF THE ADJACENT SIDE TO THE HYPOTENUSE
OR NEGATIVE SQUARE ROOT 3 DIVIDED BY 2.
NOW FOR THE LAST EXAMPLE, WE HAVE THE SINE OF 315 DEGREES.
SO AGAIN, WE'LL FIRST SKETCH THE ANGLE IN STANDARD POSITION,
SO HERE'S THE INITIAL SIDE.
3/4 OF A ROTATION COUNTER CLOCKWISE WOULD BE TO HERE.
THAT WOULD BE 270 DEGREES WHICH MEANS TO REACH 315 DEGREES,
WE NEED TO ROTATE 45 MORE DEGREES.
SO THE TERMINAL SIDE OF 315 DEGREES WOULD BE HERE.
AND SINCE ONE COMPLETE ROTATION COUNTER CLOCKWISE
WOULD BE 360 DEGREES,
THIS LEAVES US WITH A REFERENCE ANGLE OF 45 DEGREES HERE.
AND NOW, WE'LL FORM THE REFERENCE TRIANGLE.
AGAIN, WE'LL SKETCH A PERPENDICULAR SEGMENT
TO THE X AXIS.
SO NOW, WE HAVE A 45-45-90 REFERENCE TRIANGLE
WHICH MEANS WE CAN LABEL THE 2 LEGS LENGTH 1
AND THE HYPOTENUSE SQUARE ROOT 2.
AND NOW, WE'LL CHECK THE SINES OF THE LEGS.
NOTICE NOW, WE'RE IN QUADRANT 4
WHERE THE X COORDINATE IS POSITIVE
AND THE Y COORDINATE IS NEGATIVE,
SO THIS LEG HERE MUST BE -1.
SO TO FIND THE SINE OF 315 DEGREES
USING A REFERENCE TRIANGLE,
WE'LL TAKE THE RATIO OF THE OPPOSITE SIDE
TO THE HYPOTENUSE OR -1 DIVIDED BY SQUARE ROOT 2.
THIS IS THE EXACT VALUE FOR SINE 315 DEGREES.
BUT OFTEN WE'RE ASKED TO RATIONALIZE THE DENOMINATOR,
SO LET'S GO AHEAD AND SHOW HOW WE WOULD DO THAT.
WE WOULD MULTIPLY BY THE SQUARE ROOT OF 2/THE SQUARE ROOT OF 2.
SO WE HAVE NEGATIVE SQUARE ROOT 2 ALL/SQUARE ROOT 2
x SQUARE ROOT 2 = 2.
SO WE CAN EXPRESS THE SINE OF 315 DEGREES THIS WAY
IF WE'RE NOT REQUIRED TO RATIONALIZE THE DENOMINATOR,
OR THIS WAY IF WE ARE.
AND BEFORE WE GO, I'LL SHOW YOU SOME QUICK NOTES
ON THE 30-60-90 REFERENCE TRIANGLE
AND THE 45-45-90 REFERENCE TRIANGLE.
HERE'S THE 30-60-90 RIGHT TRIANGLE THAT WE USED.
WE OFTEN USE THIS TRIANGLE HERE.
BUT OF COURSE, WE COULD MULTIPLY EACH SIDE BY A CONSTANT.
IF THAT CONSTANT WAS X,
WE COULD THIS AS REFERENCE TRIANGLE AS WELL.
AND FOR THE 45-45-90 REFERENCE TRIANGLE,
WE USE THIS TRIANGLE HERE.
BUT AGAIN, WE COULD MULTIPLY EACH SIDE BY A CONSTANT.
AND AGAIN IF THAT CONSTANT WAS X,
WE WOULD HAVE THIS REFERENCE TRIANGLE HERE.
SO YOU SHOULD HAVE THESE REFERENCE TRIANGLES MEMORIZED.
BUT IF YOU DON'T, I WOULD DEFINITELY
INCLUDE THESE IN YOUR NOTES.