Ex - Sine and cosine values using the unit circle - Multiples of 30 degrees

- WE WANT TO DETERMINE THE EXACT TRIG FUNCTION VALUES USING THE UNIT CIRCLE. IF WE SKETCH THESE ANGLES IN STANDARD POSITION THE POINT WHERE THE TERMINAL SIDE INTERSECTS THE UNIT CIRCLE WILL GIVE US BOTH THE COSINE FUNCTION VALUE AND THE SINE FUNCTION VALUE. THE X COORDINATE IS EQUAL TO COSINE THETA AND THE Y COORDINATE IS EQUAL TO SINE THETA ON THE UNIT CIRCLE. SO TO FIND THE EXACT VALUE OF COSINE 150 DEGREES AND SINE 150 DEGREES WE FIRST WANT TO SKETCH 150 DEGREES IN STANDARD POSITION. SO THE INITIAL SIDE WILL BE ALONG THE X AXIS HERE AND NOW WE'LL ROTATE COUNTERCLOCKWISE 150 DEGREES. SO ROTATING TO THE Y AXIS WOULD BE 90 DEGREES SO WE NEED TO ROTATE 60 DEGREES, MORE WHICH WOULD BE TO HERE. SO HERE'S THE TERMINAL SIDE OF 150 DEGREES AND THE TERMINAL SIDE INTERSECTS THE UNIT CIRCLE AT THIS POINT HERE WHERE THE X COORDINATE IS EQUAL TO -SQUARE ROOT 3 DIVIDED BY 2 AND THE Y COORDINATE IS EQUAL TO 1/2. SO COSINE 150 DEGREES IS EQUAL TO THE X COORDINATE OR NEGATIVE SQUARE ROOT 3 DIVIDED BY 2 AND SINE 150 DEGREES IS EQUAL TO THE Y COORDINATE WHICH IS EQUAL TO 1/2. NOW LETS FIND COSINE AND SINE OF -240 DEGREES. AGAIN, THE FIRST STEP IS TO SKETCH THE ANGLE IN STANDARD POSITION SO THE INITIAL SIDE IS ALONG THE X AXIS BUT NOW BECAUSE THE ANGLE IS NEGATIVE WE'RE GOING TO ROTATE CLOCKWISE 240 DEGREES. WELL HALF A ROTATION CLOCKWISE WOULD BE -180 DEGREES AND SINCE WE WANT -240 DEGREES, WE NEED TO ROTATE 60 DEGREES, MORE WHICH WOULD BE TO HERE. NOTICE THAT -240 DEGREES IS COTERMINAL WITH 120 DEGREES.   THE TERMINAL SIDE INTERSECTS THE UNIT CIRCLE AT THIS POINT HERE WHERE THE X COORDINATE IS EQUAL TO -1/2 AND THE Y COORDINATE IS EQUAL TO SQUARE ROOT 3 DIVIDED BY 2. SO COSINE -240 DEGREES IS EQUAL TO THE X COORDINATE OF -1/2 AND SINE -240 DEGREES IS EQUAL TO THE Y COORDINATE WHICH IS SQUARE ROOT 3 DIVIDED BY 2. THERE IS ONE MORE THING TO KEEP IN MIND WHEN USING THE UNIT CIRCLE. TO FIND THE COSINE AND SINE OF -240 DEGREES WE COULD HAVE USED A COTERMINAL ANGLE. FOR EXAMPLE WE COULD HAVE ADDED 360 DEGREES TO -240 DEGREES THAT WOULD HAVE GIVEN US 120 DEGREES AND NOTICE ON THE UNIT CIRCLE 120 DEGREES IS LABELED HERE WHICH HAS THE SAME TERMINAL SIDE AS -240 DEGREES. SO THE COSINE OF 120 DEGREES AND THE SINE OF 120 DEGREES WOULD OBVIOUSLY GIVE US THE SAME FUNCTION VALUES. SO WE DO HAVE THE OPTION OF FINDING COTERMINAL ANGLES IF WE WISH. I HOPE YOU FOUND THIS HELPFUL. WE'LL TAKE A LOOK AT TWO MORE EXAMPLES IN THE NEXT VIDEO.