Determining Trig Function Values Using Reference Angles and Reference Triangles

Determining Trig Function Values Using Reference Angles and Reference Triangles


– WELCOME TO A VIDEO ON THE
TRIGONOMETRIC FUNCTION VALUES OF COMMON ANGLES. THE GOAL OF THE VIDEO
IS TO DETERMINE THE VALUE OF THE SIX TRIGONOMETRIC
FUNCTIONS USING REFERENCE ANGLES
AND REFERENCE TRIANGLES. SO THE FIRST THING
WE NEED TO REMEMBER IS THAT IF WE HAVE A 30, 60, 90
RIGHT TRIANGLE, THE RELATIONSHIP
AMONG THE 3 SIDES WOULD BE 1 FOR THE SHORTEST LEG,
2 FOR THE HYPOTENUSE, AND SQUARE ROOT OF 3
FOR THE OTHER LEG. AND OF COURSE IT COULD BE
ANY MULTIPLE OF THESE, BUT THIS IS THE MOST COMMON WAY
TO REPRESENT THE RELATIONSHIP AMONG THE 3 SIDES OF A 30, 60,
90 RIGHT TRIANGLE. NOW, IF WE TALK ABOUT
A 45, 45 RIGHT TRIANGLE. THE RELATIONSHIP AMONG THE
3 SIDES WOULD BE THE 2 LEGS WOULD BE 1 AND THE HYPOTENUSE
WOULD BE THE SQUARE ROOT OF 2 OR ANY MULTIPLE. SO THIS WAS COVERED
IN A PREVIOUS VIDEO. IF YOU DON’T KNOW THIS, YOU
MIGHT WANT TO WATCH THAT FIRST. LET’S CONSIDER THE ANGLE 120
DEGREES IN STANDARD POSITION. SO WE HAVE THIS MUCH ROTATION,
OR 30 DEGREES PASSED 90 DEGREES. HERE’S OUR NEUTRAL SIDE,
HERE’S OUR TERMINAL SIDE. THE 20 DEGREE ANGLE
IS SOMETIMES CALLED THETA. NOW, THE REFERENCE ANGLE
IS FORMED BY THE TERMINAL SIDE
OF THE ANGLE AND THE CLOSEST PART
OF THE X AXIS. SO WHAT THAT MEANS IS, PICK ANY POINT ON THE TERMINAL
SIDE OF THE ANGLE AND DRAW A SEGMENT TO THE
CLOSEST PART OF THE X AXIS AND THIS WILL FORM
A RIGHT TRIANGLE, WHICH IS OUR REFERENCE TRIANGLE. AND OUR REFERENCE ANGLE
WOULD BE THIS ANGLE HERE, SOMETIMES CALLED THETA PRIME. SO THE TRIANGLE FORMED
BY THETA PRIME CAN BE USED TO FIND THE VALUES OF THE SIX TRIGONOMETRIC
FUNCTION VALUES OF THETA, OR IN THIS CASE, 120 DEGREES. SO LETS TAKE A CLOSER LOOK
AT THIS. LET’S SEE IF WE CAN DETERMINE
THE SIX TRIGONOMETRIC VALUES FOR 120 DEGREES. SO GO AHEAD
AND LETS SET THIS UP. WE KNOW THIS IS 120 DEGREES. WE’LL FORM OUR REFERENCE
TRIANGLE BY THIS SEGMENT HERE TO THE CLOSEST PART
OF THE X AXIS. HERE’S OUR REFERENCE ANGLE,
WHICH IF THIS IS 120, WE KNOW THIS WOULD HAVE TO BE
60 DEGREES. SO IF THIS IS 60 DEGREES, THIS OF COURSE IS 30 DEGREES AND NOW WE KNOW HOW THESE 3
SIDES RELATE TO ONE ANOTHER. THIS WOULD BE ONE,
THIS WOULD BE TWO, AND THIS WOULD BE
THE SQUARE ROOT OF THREE. NOW, THE ONLY CATCH HERE IS,
NOTICE, THAT THE X COORDINATE OF THIS
POINT WOULD ACTUALLY BE -1. SO WE ARE GOING TO HAVE TO LABEL
THIS AS A -1. A Y COORDINATE WOULD BE
+SQUARE ROOT THREE SO WE LEAVE THAT ALONE. WE CAN USE THIS
REFERENCE TRIANGLE TO FIND THE SIX TRIGONOMETRIC
FUNCTION VALUES. SO USING SOHCAHTOA,
IF WE NEED TO, THE SINE OF 120 WOULD BE THE
RATIO OF THE OPPOSITE SIDE TO THE HYPOTENUSE
OR SQUARE ROOT 3/2, WHICH MEANS THE COSECANT
OF 120 DEGREES WOULD BE THE RECIPROCAL OF THIS. REMEMBER THAT SINE THETA AND
COSECANT THETA ARE RECIPROCALS. WE MAY BE ASKED
TO RATIONALIZE THIS, WHICH WOULD GIVE US
TWO SQUARE ROOT THREE/THREE. THE COSINE OF 120 DEGREES WOULD BE THE COSINE
OF THIS GREEN ANGLE HERE, WHICH IS ADJACENT
OVER HYPOTENUSE -1/2. AND SINCE SECANT
IS A RECIPROCAL OF COSINE, THIS WOULD GIVE US -2/1 OR -2. AND FOR THE TANGENT
OF 120 DEGREES IT WOULD BE THE RATIO
OF THE OPPOSITE SIDE TO THE ADJACENT SIDE, OR SQUARE ROOT 3 DIVIDED BY -1
=-SQUARE ROOT 3. AND THE RECIPROCAL
FOR A COTANGENT 120 DEGREES WOULD BE -1 SQUARE ROOT 3. AND THAT’S IT. WE DON’T NEED A CALCULATOR
TO DETERMINE THESE VALUES. AND NOT ONLY THAT,
WE’RE FINDING EXACT VALUES WHETHER IT’S RATIONAL
OR IRRATIONAL. LET’S TRY ANOTHER ONE. LET’S PLOT 210 DEGREES
IN STANDARD POSITION AND THEN FIND THE VALUES
OF THESE. HERE’S OUR INITIAL SIDE, WE’LL ROTATE COUNTERCLOCKWISE
210 DEGREES. SO THIS WOULD BE 180 AND 30 DEGREES MORE
TO OUR TERMINAL SIDE HERE. PICK ANY POINT
ON THE TERMINAL SIDE. CONNECT IT TO THE CLOSEST PART
OF THE X AXIS. THIS WOULD BE OUR
REFERENCE TRIANGLE. THIS IS OUR REFERENCE ANGLE,
WHICH IF THIS IS 180. THIS REFERENCE ANGLE WOULD BE
30 DEGREES. SO WE HAVE ANOTHER 30, 60, 90
RIGHT TRIANGLE. SO WE KNOW THAT THE SHORT LEG
WOULD BE SUM MULTIPLE OF ONE, THE HYPOTENUSE WOULD BE
SUM MULTIPLE OF TWO, AND THE OTHER LEG
WOULD BE SUM MULTIPLE OF THE SQUARE ROOT OF THREE. AGAIN, THE ONLY CONDITION HERE
WE HAVE TO BE CAREFUL ABOUT IS THIS COORDINATE HERE WOULD HAVE BOTH A -X AND -Y
COORDINATE, SO THE X VALUE WOULD HAVE TO BE
NEGATIVE AND SO WOULD THE Y. SO THE SINE OF 210 DEGREES
WOULD BE THE RATIO OF THE OPPOSITE SIDE TO THE
HYPOTENUSE OR -1/2, -1/2. COSECANT WOULD BE THE RECIPROCAL
A -2. THE COSINE OF 210 WOULD BE
THE RATIO OF THE ADJACENT SIDE TO THE HYPOTENUSE
OR -SQUARE ROOT THREE/TWO. AND THEN THE SECANT WOULD BE
THE RECIPROCAL. IF WE RATIONALIZE THIS SIMILAR
TO THE PREVIOUS SLIDE, WE WOULD HAVE -2 SQUARE ROOT
3/3. TANGENT OF 2/10
WOULD BE THE RATIO OF THE OPPOSITE SIDE TO THE
ADJACENT SIDE -1/-SQUARE ROOT 3. SO IF WE WERE REQUIRED
TO RATIONALIZE THIS, IT WOULD BE THE SQUARE ROOT
OF THREE/THREE. AND THEN OF COURSE, THE COTANGENT WOULD BE
THE RECIPROCAL OF THIS, WHICH IS THE SQUARE ROOT
OF THREE. LET’S TRY ANOTHER. NOW WE WANT TO PLOT -45 DEGREES,
SO WE HAVE INITIAL SIDE HERE, ROTATION CLOCKWISE
OF 45 DEGREES, SKETCH THE TERMINAL SIDE,
FORM THE REFERENCE TRIANGLE. AND SO THIS WOULD BE 45 DEGREES,
THIS WOULD ALSO BE 45 DEGREES, WHICH MEANS WE SHOULD KNOW
THE RELATIONSHIP AMONG THESE THREE SIDES AND IT WOULD BE ONE, ONE,
SQUARE ROOT TWO. AGAIN, WATCHING THE SIGNS, NOW WE’RE
IN THE SECOND QUADRANT. SO THE X COORDINATE WOULD BE
POSITIVE, BUT THE Y COORDINATE
WOULD BE NEGATIVE. SO WE’LL CALL THIS -1. WE’RE READY TO GO. WE’RE GOING TO USE THIS
REFERENCE ANGLE HERE TO FIND ALL OF THESE VALUES. SINE OF -45 DEGREES WOULD BE
THE RATIO OF THE OPPOSITE SIDE TO THE HYPOTENUSE. -1/SQUARE ROOT 2, RATIONALIZING
THIS WE’D HAVE -SQUARE ROOT 2/2 AND THE COSECANT WOULD BE
THE RECIPROCAL OF THIS IN BLUE. SO JUST -SQUARE ROOT TWO. COSINE WOULD BE THE ADJACENT
OVER THE HYPOTENUSE. +1/THE SQUARE ROOT OF 2,
RATIONALIZED SQUARE ROOT 2/2. COSECANT WOULD AGAIN,
WOULD BE THE RECIPROCAL OF THE COSINE RATIO
SQUARE ROOT TWO. AND TANGENT OF OUR ANGLE
WOULD BE THE OPPOSITE OVER THE ADJACENT -1/+1,
WHICH EQUALS -1. AND THE RECIPROCAL
WOULD STILL BE -1. AND IT LOOKS LIKE
I HAVE ONE MORE, SO LET’S GO AHEAD AND TAKE A
LOOK AT THE ANGLE 270 DEGREES. HERE’S OUR INITIAL SIDE. NOW, THIS ONE WOULD BE
90, 180, 270, SO THIS IS A QUADRANTAL ANGLE. SO WE’RE NOT ABLE TO FORM
A REFERENCE TRIANGLE, BUT WHAT WE COULD THINK OF
IS THE UNIT CIRCLE. REMEMBER THAT THE UNIT CIRCLE
WOULD BE, IF WE HAD A CIRCLE ON HERE WHERE THE TERMINAL SIDE
INTERSECTED IT WITH RADIUS ONE. SO IF WE COULD DETERMINE
THE COORDINATES OF THIS POINT, WE COULD USE THIS TO FIND
THE VALUES OF THESE FUNCTIONS. AND OF COURSE,
WE COULD USE THE– AND THIS WOULD HAVE
THE COORDINATES (0, -1). SO IN THIS CASE WE WOULD THINK
OF THIS AS X, THIS AS Y, AND REMEMBER IT’S A UNIT CIRCLE,
SO R IS EQUAL TO ONE. SO JUST TO REFRESH YOUR MEMORY,
SINE THETA IS EQUAL TO Y/R. SO HERE WE’D HAVE -1/1,
WHICH IS EQUAL TO -1. THE RECIPROCAL IS ALSO A -1. COSINE THETA IS X/R,
WHICH WOULD BE X/1 OR JUST X, WHICH IS EQUAL TO 0. REMEMBER THE RECIPROCAL OF ZERO
WOULD BE UNDEFINED. AND THE TANGENT THETA IS Y/X. -1/0 IS ALSO UNDEFINED. BUT WRITING THIS FIRST
IS HELPFUL BECAUSE IF YOU TAKE THE
RECIPROCAL OF THIS FOR COTANGENT WE’D HAVE 0/-1, WHICH IS EQUAL TO 0. LASTLY,
I’LL SHOW THE UNIT CIRCLE, THE POINTS THAT WOULD FORM ALL OF THE MOST COMMON
REFERENCE ANGLES, MEANING 30, 45, 60
IN ALL 4 QUADRANTS. SO YOU PROBABLY SHOULD HAVE
ONE OF THESE HANDY BECAUSE IT CAN SPEED UP
THE PROCESS TO FIND THE TRIGONOMETRIC
FUNCTION VALUES OF THE MOST COMMON ANGLES. WE WENT OVER QUITE A BIT
OF INFORMATION HERE, BUT HOPEFULLY
YOU FOUND IT HELPFUL. HAVE A GOOD DAY.  

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