– 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.

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!

I think there is a typo at 4:44. The bottom half should say ODD Trigonometric Identities.

Can you just be my math teacher? Please?

thanks, you saved me, i wasnt paying attention in class D:

Exactly what I was going to say.

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

thanks so much

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.

Your videos are always very informative and helpful.

You're the best in the world

Excellent video! Thank you so much!

my name is jeff