Even and Odd Trigonometric Identities

Even and Odd Trigonometric Identities


– WELCOME TO A LESSON ON EVEN AND ODD TRIGONOMETRIC
IDENTITIES. THE GOAL OF THE VIDEO
IS TO STATE AND ILLUSTRATE THE EVEN AND ODD
TRIG IDENTITIES. LET’S START BY TALKING
ABOUT EVEN FUNCTIONS. EVEN FUNCTIONS ARE SYMMETRICAL
ACROSS THE Y AXIS. HERE ARE TWO FUNCTIONS
THAT ARE EVEN. IF WE WERE TO FOLD THESE TWO
FUNCTIONS ACROSS THE Y AXIS, THE RIGHT HALF WOULD MATCH UP
PERFECTLY WITH THE LEFT HALF, MEANING IF WE FOLDED THIS
ACROSS THE Y AXIS, THIS HALF HERE WOULD FALL
DIRECTLY ON THIS HALF HERE. ANOTHER WAY TO THINK OF IT IS THAT EVERY POINT ON THE
RIGHT SIDE OF THIS FUNCTION– LET’S SAY THIS POINT HERE, HAS A MIRROR IMAGE ON THE LEFT
SIDE, THIS POINT RIGHT HERE. SO WE CAN SAY THIS FUNCTION
IS A MIRROR IMAGE ACROSS THE LINE Y=X, AND THE SAME IS TRUE FOR
THE GRAPH OF Y=COSINE THETA. IF WE WERE TO FOLD THIS GRAPH
ACROSS THE Y AXIS, IT WOULD MATCH UP PERFECTLY
WITH THE OTHER HALF. THIS HALF HERE WOULD MATCH UP
PERFECTLY WITH THIS HALF HERE. MORE SPECIFICALLY, IF WE TAKE
A LOOK AT THIS FUNCTION HERE, F OF 2=2. IF WE CHANGE THE SIGN
OF THE X COORDINATE, LET’S SAY WE WANT F OF -2, NOTICE THAT F OF -2
IS ALSO EQUAL TO +2. SO IF THE FUNCTION IS EVEN, IF WE CHANGE THE SIGN
OF THE X COORDINATE, THE Y COORDINATE
OR THE FUNCTION VALUE REMAINS THE SAME. TO GENERALIZE THIS,
WE SAY F OF X=F OF -X IF THE FUNCTION IS EVEN. FOR THE TRIG FUNCTION,
IF WE CONSIDER X=PI/2, THIS POINT HERE,
WE KNOW F OF PI/2=0, AND F OF -PI/2 ALSO=0. SO HERE ARE THE EVEN
TRIG IDENTITIES. COSINE OF NEGATIVE THETA
=COSINE THETA, AND THE SAME IS TRUE FOR THE RECIPROCAL FUNCTION
SECANT THETA. AND AGAIN, YOU CAN SEE
GRAPHICALLY THAT THESE ARE SYMMETRICAL
ACROSS THE Y AXIS. NOW, LET’S TALK ABOUT
ODD FUNCTIONS. ODD FUNCTIONS ARE SYMMETRICAL
ABOUT THE ORIGIN, WHICH MEANS THEY HAVE
ROTATIONAL SYMMETRY ABOUT THE ORIGIN, WHICH MEANS IF WE ROTATE
THIS 180 ABOUT THE ORIGIN, THE FUNCTION WOULD LOOK
EXACTLY THE SAME. THIS BLUE HALF AND THIS GREEN
HALF WOULD JUST SWITCH PLACES IF WE ROTATED THIS 180
IN EITHER DIRECTION ABOUT THE ORIGIN. AND THE SAME IS TRUE FOR
THE GRAPH OF Y=SINE THETA. IF WE ROTATE THIS
ABOUT THE ORIGIN 180 , THIS BLUE HALF AND THIS GREEN
HALF WOULD JUST SWITCH PLACES. SO LET’S SEE WHAT HAPPENS WHEN WE CONSIDER FUNCTION
VALUES ON ODD FUNCTIONS. NOTICE IN THIS GRAPH,
F OF 2=+2, BUT F OF -2, IF WE SWITCH THE SIGN
OF THE X COORDINATE, NOTICE HOW THE Y COORDINATE
ALSO SWITCHES SIGNS. F OF -2=-2, SO TO STATE
THIS RELATIONSHIP IN GENERAL, WE SAY THAT F OF X
=THE OPPOSITE OF F OF -X OR IF WE MULTIPLY
BOTH OF THESE BY -1 AND THEN FLIP IT AROUND, WE CAN SAY THAT F OF -X
=THE OPPOSITE OF F OF X, MEANING IF THE FUNCTION IS ODD AND WE CHANGE THE SIGN
OF THE X COORDINATES, THE Y COORDINATES OR THE FUNCTION VALUES
WILL BE THE OPPOSITE SIGN. FOR THE TRIG FUNCTION,
NOTICE THAT F OF PI/2=1, AND F OF -PI/2=-1. SO LET’S TAKE A LOOK
AT THE ODD TRIG IDENTITIES. HERE THEY ARE FOR SINE
AND COSECANT, BECAUSE THESE ARE
ODD FUNCTIONS, AND THE SAME IS TRUE
FOR TANGENT AND COTANGENT. TANGENT AND COTANGENT
ARE ODD FUNCTIONS. THEREFORE, THEY HAVE
ROTATIONAL SYMMETRY ABOUT THE ORIGIN, AND HERE’S A SUMMARY
OF THOSE IDENTITIES. LET’S TAKE A LOOK
AT SOME EXAMPLES. 1st EXAMPLE, IF SINE X=0.75, THEN SINE OF -X
IS GOING TO BE=-0.75. THE SINE FUNCTION
IS AN ODD FUNCTION. SO IF WE CHANGE THE SIGN
OF THE X COORDINATE, THE FUNCTION VALUES
WILL BE THE OPPOSITE SIGN, BUT THE COSINE FUNCTION
IS AN EVEN FUNCTION. SO IF WE CHANGE THE SIGN
OF THE X COORDINATES ON THE SINE FUNCTION, THE FUNCTION VALUES
STAY THE SAME. SO IF COSINE X=0.2,
THEN COSINE -X ALSO=0.2. THE TANGENT FUNCTION
IS AN ODD FUNCTION. SO IF TANGENT X=5.3,
THEN TANGENT -X=-5.3. AND OUR LAST EXAMPLE, THE SECANT FUNCTION
IS AN EVEN FUNCTION. IF SECANT -X=2.9,
THEN SECANT X=2.9 AS WELL. I HOPE YOU FOUND
THESE EXPLANATIONS HELPFUL. THANK YOU FOR WATCHING.  

11 thoughts on “Even and Odd Trigonometric Identities

  1. thanks..your examples in the end really helped..as i have never used these identities before, so wouldnt know what to apply even if I saw a problem for it..your examples in the end really helped! I'm sure it will be on my test tomorrow!

  2. Exactly what I was going to say.

    Hope no one gets confused about it. Doubt it though, since it's presented so well.

  3. Thank you so very much.  My Trig book has no explanation of what the even & odd identities mean, so memorizing the words was doing me no good.  Your simple explanation has made a lot of difference.   Your video is very much appreciated. 

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