- IF COSINE THETA EQUALS 5/13 AND THETA IS IN QUADRANT FOUR,
WE WANT TO DETERMINE EACH TRIG FUNCTION VALUE.
SO TO FIND THESE TRIG FUNCTION VALUES,
WE'RE GOING TO SKETCH THE REFERENCE TRIANGLE
FOR ANGLE THETA.
AND SINCE THETA IS IN QUADRANT FOUR, OR THIS QUADRANT HERE,
LET'S GO AHEAD AND ASSUME
THAT THIS IS THE TERMINAL SIDE OF ANGLE THETA,
WHICH MEANS THE REFERENCE ANGLE WOULD BE THE ANGLE
BETWEEN THE X-AXIS AND THIS TERMINAL SIDE
OR THIS ANGLE HERE.
SO WE'LL CALL THIS THETA SUB-R AND THE REFERENCE TRIANGLE
WOULD BE FORMED BY SKETCHING A SEGMENT FROM THE TERMINAL SIDE
THAT'S PERPENDICULAR TO THE X-AXIS OR A SEGMENT LIKE THIS.
SO NOW WE HAVE OUR REFERENCE ANGLE
AND WE HAVE THE REFERENCE TRIANGLE,
WHICH I'LL OUTLINE IN BLACK.
SO IF COSINE THETA IS EQUAL TO 5/13,
THAT MEANS THE RATIO OF THE ADJACENT SIDE
TO THE HYPOTENUSE MUST BE 5/13.
SO WE CAN LABEL THIS LEG OF THE REFERENCE TRIANGLE 5
AND THE HYPOTENUSE 13.
AN IMPORTANT THING TO REMEMBER ABOUT REFERENCE TRIANGLES
IS THE LEGS CAN BE POSITIVE OR NEGATIVE,
BUT THE HYPOTENUSE IS ALWAYS POSITIVE.
SO NOW WHEN WE GO TO FIND THE LENGTH OF THIS LEG HERE,
WHICH WE'LL LABEL Y,
SINCE WE'RE IN THE FOURTH QUADRANT
WHERE X IS POSITIVE AND Y IS NEGATIVE,
WE'LL LABEL THIS LEG AS A NEGATIVE.
NOW WE SHOULD RECOGNIZE THIS RIGHT TRIANGLE
AS A 5, 12, 13 RIGHT TRIANGLE.
BUT JUST IN CASE WE DON'T,
WE CAN FIND Y USING THE PYTHAGOREAN THEOREM.
WE WOULD HAVE Y SQUARED PLUS 5 SQUARED MUST EQUAL 13 SQUARED.
SO WE'D HAVE Y SQUARED PLUS 25 EQUALS 169.
SUBTRACT 25 ON BOTH SIDES.
WE HAVE Y SQUARED EQUALS 144.
AND NOW SQUARE ROOT BOTH SIDES OF THE EQUATION.
REMEMBER, Y CAN BE BOTH POSITIVE OR NEGATIVE,
SO WE HAVE Y EQUALS PLUS OR MINUS 12.
AND, AGAIN, BECAUSE WE'RE IN THE FOURTH QUADRANT
WHERE THE Y COORDINATE IS NEGATIVE,
WE ARE GOING TO LABEL THIS LEG -12.
SO FOR THIS POINT ON THE COORDINATE PLANE,
THE X-COORDINATE WOULD BE 5 AND THE Y-COORDINATE WOULD BE -12.
BUT NOW THAT WE HAVE OUR REFERENCE TRIANGLE,
WE CAN EASILY FIND THESE FIVE TRIG FUNCTION VALUES.
SINE THETA IS EQUAL TO THE RATIO
OF THE OPPOSITE SIDE TO THE HYPOTENUSE
OR -12 DIVIDED BY 13, OR -12/13.
THERE'S TWO WAYS TO FIND SECANT THETA.
SINCE SECANT IS EQUAL TO THE RECIPROCAL OF COSINE
AND WE'RE TOLD COSINE THETA EQUALS 5/13,
THEN SECANT THETA IS EQUAL TO THE RECIPROCAL OR 13/5.
USING THE REFERENCE TRIANGLE,
WE WOULD USE THE RATIO OF THE HYPOTENUSE
TO THE ADJACENT SIDE OR 13/5, WHICH WE ALREADY HAVE.
AND THERE'S TWO WAYS TO FIND COSECANT THETA AS WELL.
SINCE COSECANT IS EQUAL TO THE RECIPROCAL OF SINE,
AND WE KNOW SINE THETA EQUALS -12/13,
THEN THE COSECANT THETA IS EQUAL TO -13/12.
OR USING OUR REFERENCE TRIANGLE,
IT HAD THE RATIO OF THE HYPOTENUSE TO THE OPPOSITE SIDE.
AGAIN, -12/13.
TANGENT THETA IS EQUAL TO THE RATIO
OF THE OPPOSITE SIDE TO THE ADJACENT SIDE,
WHICH WOULD BE -12 DIVIDED BY 5, OR -12/5.
AND THEN AGAIN, FOR COTANGENT THETA,
AND SINCE COTANGENT IS EQUAL TO THE RECIPROCAL OF TANGENT,
IF TANGENT THETA IS EQUAL TO -12/5,
THEN COTANGENT THETA IS EQUAL TO -5/12.
OR USING THE REFERENCE TRIANGLE,
WE'D HAVE THE RATIO OF THE ADJACENT SIDE
TO THE OPPOSITE SIDE.
5 DIVIDED BY -12 IS -5/12.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.