- WELCOME TO A VIDEO ON DETERMINING
THE TRIGONOMETRIC FUNCTION VALUES USING THE UNIT CIRCLE.
I WOULD RECOMMEND THAT YOU WATCH THE VIDEO
ON HOW TO FIND TRIG FUNCTION VALUES USING REFERENCE ANGLES
AND REFERENCE TRIANGLES FIRST.
SO BEFORE WE JUMP TO THE UNIT CIRCLE LETS TAKE A LOOK
AT OUR REFERENCE TRIANGLES.
REMEMBER IF WE HAVE A 30-60-90 RIGHT TRIANGLE,
THE SIDES OF THE RIGHT TRIANGLE RELATE IN A SPECIAL WAY.
WE COULD LABEL THE SHORT LEG 1, THE HYPOTHENUSE 2,
AND THE OTHER LEG SQUARE ROOT OF 3 OR ANY MULTIPLE OF THIS
AND FOR THE 45-45 RIGHT TRIANGLE WE COULD LABEL THE TWO LEGS 1
AND THE HYPOTHENUSE SQUARE ROOT 2.
NOW JUST TO REFRESH YOUR MEMORY WHEN WE HAVE THE UNIT CIRCLE
WHAT THAT MEANS IS THE RADIUS OF THIS CIRCLE IS EQUAL TO 1.
SO IF WE SKETCH AN ANGLE IN STANDARD POSITION
AND SKETCHED OUR REFERENCE TRIANGLE
THE HYPOTHENUSE OF THIS TRIANGLE WOULD EQUAL 1 UNIT.
SO GOING BACK UP TO OUR REFERENCE TRIANGLES
IF WE WERE TO TAKE THE HYPOTHENUSE OF THIS TRIANGLE
AND MAKE IT EQUAL TO 1 BY DIVIDING BY 2
WE CAN DO THIS AS LONG AS WE DIVIDE EVERYTHING BY 2.
OF COURSE 2 DIVIDED BY 2 = 1.
THIS WOULD BE SQUARE ROOT 3 DIVIDED BY 2
AND THIS WOULD EQUAL 1/2.
THESE ARE THE VALUES THAT WE'LL SEE ON THE UNIT CIRCLE
BECAUSE THE HYPOTHENUSE OF THE REFERENCE TRIANGLE
CORRESPONDS TO THE RADIUS OF THE UNIT CIRCLE.
ON THIS OTHER RIGHT TRIANGLE
IF WE WANT THE HYPOTHENUSE TO EQUAL 1
WE'D HAVE TO DIVIDE BY THE SQUARE ROOT OF 2
AND WHAT WE'LL SEE NOW IS THE HYPOTHENUSE IS EQUAL TO 1.
IF WE RATIONALIZE THIS,
WE WOULD HAVE THE SQUARE ROOT OF 2 DIVIDED BY 2
IN BOTH CASES AND THESE ARE THE VALUES
THAT WE'LL ALSO SEE ON THE UNIT CIRCLE.
SO LET'S GO AHEAD AND TAKE A LOOK AT THE UNIT CIRCLE.
WHAT WE CAN SEE ON THIS UNIT CIRCLE
ARE THE LABELS OF THE MOST COMMON ANGLES
IN STANDARD POSITION FOR US.
KEEP IN MIND THAT THE RADIUS R = 1
SO WHAT THAT TELLS US IS THAT THE X COORDINATE
IS EQUAL TO COSINE THETA
AND THE Y COORDINATE IS EQUAL TO SINE THETA
AND OF COURSE TANGENT THETA WOULD EQUAL Y/X.
SO LETS GO AHEAD AND USE THIS TO FIND SOME TRIG FUNCTION VALUES.
SO WHAT WE HAVE TO FIGURE OUT
IS WHERE DOES THE TERMINAL SIDE
OF THE ANGLE IN STANDARD POSITION
INTERSECT THE UNIT CIRCLE.
LET'S GO AHEAD AND SKETCH THIS ANGLE AND SEE.
BECAUSE IT'S NEGATIVE WE'LL GO CLOCKWISE 150 DEGREES
SO THERE'S -90 AND -60 MORE WOULD BE HERE
SO THIS IS THE POINT ON THE UNIT CIRCLE
THAT WE CAN USE TO DETERMINE THE SINE OF -150.
REMEMBER SINE IS EQUAL TO Y SO WE HAVE -1/2.
LET'S TAKE A LOOK AT THE COSINE OF 960 DEGREES.
FOR THIS ANGLE I'M GOING TO GO AHEAD
AND FIND A SMALLER COTERMINAL ANGLE
SO IF I SUBTRACT 720 DEGREES FROM THIS
REMEMBER THAT'S LIKE TWO REVOLUTIONS
I WOULD HAVE THE COSINE OF 240 DEGREES
WHICH WOULD EQUAL THE COSINE OF 960 DEGREES.
LET'S GO AHEAD AND SKETCH THIS.
SO FROM HERE TO HERE WOULD BE 180
AND WE NEED TO ROTATE 60 DEGREES MORE TO HERE.
THE TERMINAL SIDE INTERSECTS THE UNIT CIRCLE AT THIS POINT.
REMEMBER THAT COSINE THETA IS EQUAL TO X
SO IT LOOKS LIKE WE HAVE -1/2 AGAIN.
NOW LETS LOOK AT TANGENT 180 DEGREES.
SO AGAIN HERE WOULD BE THE INITIAL SIDE
WE ROTATE COUNTER CLOCKWISE 180 DEGREES.
HERE'S THE TERMINAL SIDE
SO WE'LL USE THIS POINT TO DETERMINE
THE TANGENT OF 180 DEGREES.
REMEMBER TANGENT IS Y/X SO WE'D HAVE 0/-1
WHICH IS EQUAL TO ZERO.
LET'S GO AHEAD AND TAKE A LOOK AT A FEW MORE.
NOW WE WANT TO FIND THE COSECANT OF 225 DEGREES.
LET'S SKETCH IT.
WE CAN SEE IT LABELED HERE.
THERE'S THE TERMINAL SIDE SO WE CAN USE THIS POINT
TO DETERMINE THIS VALUE.
REMEMBER THAT COSECANT THETA
IS EQUAL TO THE RECIPROCAL OF SINE THETA
AND SINE THETA IS EQUAL TO Y.
SO WE WANT TO FIND THE RECIPROCAL OF -SQUARE ROOT 2/2,
WHICH WOULD BE -2/SQUARE ROOT 2.
LETS GO AHEAD AND RATIONALIZE THIS
WE'D HAVE -2 SQUARE ROOT 2 DIVIDED BY 2,
2s SIMPLIFY WE'D HAVE -SQUARE ROOT 2.
NEXT, WE HAVE THE SECANT OF -240 DEGREES.
SO WE'LL GO CLOCKWISE THERE'S -180.
WE HAVE TO GO -60 MORE TO HERE.
SO HERE'S THE TERMINAL SIDE.
WE USE THIS POINT TO DETERMINE THIS VALUE.
REMEMBER THAT SECANT THETA IS THE RECIPROCAL
OF COSINE THETA AND COSINE THETA IS EQUAL TO X ON THE UNIT CIRCLE
SO THE RECIPROCAL OF -1/2 WOULD BE -2/1,
WHICH OF COURSE EQUALS -2
AND THE LAST ONE WE HAVE COTANGENT 540 DEGREES.
SO LET'S GO AHEAD AND SKETCH THIS.
THERE'S 360 AND IF WE SUBTRACT 360 WE WOULD HAVE 180
SO WE NEED TO ROTATE ANOTHER HALF OF A REVOLUTION TO HERE.
AGAIN, WE'RE DOING THIS TO DETERMINE
WHERE THE ANGLE INTERSECTS THE UNIT CIRCLE.
WE USE THESE COORDINATES TO DETERMINE THIS VALUE.
TANGENT THETA IS Y/X SO COTANGENT
WOULD BE X/Y -1 DIVIDED BY 0,
WHICH IS UNDEFINED.
SO THAT'S HOW YOU CAN USE THE UNIT CIRCLE
TO DETERMINE TRIG FUNCTION VALUES.
I HOPE YOU FOUND THIS TUTORIAL HELPFUL.