# Examples: Determine Exact Trig Function Values With the Angle in Radians Using the Unit Circle

– WELCOME TO EXAMPLES
OF DETERMINING THE EXACT TRIG FUNCTION VALUES WHEN THE ANGLE IS GIVEN IN
RADIANS USING THE UNIT CIRCLE. IF THE ANGLE IS SKETCHED
IN STANDARD POSITION, THE POINT WHERE THE TERMINAL
SIDE OF THE ANGLE INTERSECTS A UNIT CIRCLE
WILL GIVE US THE VALUE OF COSINE THETA AND SINE THETA
WHERE THE (X) COORDINATE IS COSINE THETA AND THE (Y)
COORDINATE IS SINE THETA. SO LET’S GO AHEAD
AND SKETCH 7PI/2 RADIANS IN STANDARD POSITION. WELL, WE KNOW ONE COMPLETE
REVOLUTION COUNTER CLOCKWISE WOULD BE 2/PI RADIANS. SO LET’S START WITH THAT. THERE IS ONE REVOLUTION COUNTER
CLOCKWISE. SO IF WE HAVE 7PI/2 RADIANS
AND WE JUST ROTATED 2/PI RADIANS WHICH IS THE SAME AS 4PI/2 NOTICE THIS SIMPLIFIES
TO 2/PI RADIANS. WE’D HAVE TO ROTATE ANOTHER
3PI/2 RADIANS TO REACH 7PI/2. SO WE’RE GOING TO HAVE 3/4
OF ANOTHER REVOLUTION TO HERE FOR 7PI/2. SO THIS IS THE TERMINAL SIDE
OF OUR ANGLE. AND NOTICE THAT THE (X)
COORDINATE IS (0) AND THE (Y) COORDINATE IS (-1) WHICH TELL US THAT THE COSINE
FUNCTION VALUE WOULD BE (0) AND THE SINE FUNCTION VALUE
WOULD BE (-1). WE ALSO KNOW THAT TANGENT THETA
IT WOULD BE (Y) DIVIDED BY (X). WELL, (Y) DIVIDED BY (X)
WOULD BE (-1) DIVIDED BY (0) WHICH IS UNDEFINED WHICH TELLS
US THAT THE TANGENT FUNCTION VALUE OF DATA DOES NOT EXIST. WHILE WE’RE HERE WE KNOW
THAT COTANGENT THETA WOULD BE THE RECIPROCAL
OF TANGENT THETA OR THE RECIPROCAL OF -1/0 WOULD
BE 0/-1 WHICH IS EQUAL TO 0. NOW, LET’S GO BACK UP
TO SECANT THETA. SECANT THETA IS A RECIPROCAL
OF COSINE THETA. SO IF WE MAKE 0 INTO A FRACTION AND DETERMINE THE RECIPROCAL
THAT WILL GIVE US THE VALUE OF SECANT THETA. WELL, THE EASIEST WAY TO MAKE
SOMETHING A FRACTION IS TO PUT IT OVER 1. WELL, THE RECIPROCAL OF 0/1
WOULD BE 1/0 OR 1 DIVIDED BY 0 WHICH AGAIN IS UNDEFINED AND, THEREFORE,
SECANT THETA DOES NOT EXIST. AND THEN COSECANT THETA
IS A RECIPROCAL OF SINE THETA. SO, AGAIN, IF WE MAKE -1
INTO A FRACTION AND THEN DETERMINE
THE RECIPROCAL, THE RECIPROCAL
IS STILL EQUAL TO -1, THEREFORE,
COSECANT THETA IS EQUAL TO -1. AND NOW WE HAVE ALL SIX TRIG
FUNCTION VALUES FOR OUR ANGLE 7PI/2 RADIANS.