Question

"Show that sin[b] is an odd function and cos[b] is even, no matter what the base b is."

I'm substituting [b] for base in this instance.

Can someone explain to me how to show/prove this? I know that f(x)= -x is an odd function, and f(x)= x is an even function, but how do I go about showing something like this for sin and cos?

Answer 1

sin(-b)=-sin(b)
cos(-b)=cos(b)

Answer 2

Oh! Is that all I need to do?

Answer 3

you can write more to show why cos and sin behave like this from their definitions

Answer 4

Okay . . . would drawing the graph of sin and cos help, or would that just be pointless?

Answer 5

I'm not sure how you would generalize this using `any` b.
I mean I can't think of a way to write this as more of a proof or whatever they're looking for.

But yah using specific angles, and your unit circle,
you can do some fun stuff.

Example:\[\Large\rm \cos(60^o)=\frac{1}{2}, \qquad\qquad \cos(-60^o)=\frac{1}{2}\]\[\Large\rm \implies \qquad \cos(60^o)=\cos(-60^o)\]

Answer 6

\[\Large\rm \sin(30^o)=\frac{1}{2},\qquad\qquad\sin(-30^o)=-\frac{1}{2}\]\[\Large\rm \implies\qquad \sin(30^o)=-\sin(-30^o)\]

Answer 7

But again :P I'm not sure how we should generalize this.. hmmmmm

Answer 8

I'm pretty sure base is involved somehow, but I'm just not seeing how right now. I've been working at this for hours for several days, and I'm still stuck.

Answer 9

I don't understand what base means.
Logs have bases, trig functions do not.

What does base mean for this particular problem?

Answer 10

Well, it's kind of hard to explain it typing here, so I'll take a picture of the problem and see if that helps. My professor uses base values such as 360 degrees, 2pi radians, 400 gradians, 1 revolution, and so on. It has to do with the unit circle, and where the terminal point lies on the unit circle, and wrapping functions (as in how many times it "wraps" around the unit circle")

Answer 11

I think it is either radian or degree.

Answer 12

Maybe, but the problem wants me to show that it's true for /any/ base value . . . which I take to mean that I can use all of those values: radians, degrees, gradians, revolutions, and so on.

Answer 13

I'm a little concerned with your teacher's definition of odd and even functions.

Odd functions:\[\Large\rm f(-x) = -f(x)\]
Even functions:\[\Large\rm f(-x) = f(x)\]
Maybe he's just wording it in some strange way though.. hmm

Answer 14

we have the whole system to convert radian to degree and vice versa. They are equivalent. Really not know why your prof asks you to prove this...... weird problem.

Answer 15

Tell me about it . . . @zepdrix, I think you're right, what you put for odd/even functions is correct . . . . Thank you guys for helping me though, I really appreciate it.

Answer 16

oh base for sin and cos is something else :O

Answer 17

does it mean like different number bases?

Answer 18

I'm not sure how to explain it . . . usually the base our professor uses is paired with angles. Ex. A[base2pi](6). It has to do with wrapping functions and the unit circle.

Answer 19

is Sin_base_2pi of (2pi) = 0?

Answer 20

or is it Sin_base_10 of (2pi)=0

Answer 21

@ikram002p

Answer 22

The first one was correct. We divide the part in the parentheses by the base, kinda like this: Ex. A[base2pi](6) = 6/2pi

Answer 23

okay so the base is like, the number of radians for 1 full rotation?

Answer 24

Yes, My professor explained it like this: 1 revolution= 360 degrees= 2pi radians=400 gradians

Answer 25

use any cos / sin triagometry fact like
u can start from here
sin (-b)= sin (2pi - b)
cos (-b) = cos (2 pi -b)

Answer 26

wait why u taged me dan ?
which Qn ?

Answer 27

show this one, sin and cos odd and even respectively in all bases

Answer 28

i feel like this is trivial -'ve sign on a number works the same way, no matter what hte base you are picking

Answer 29

if positive is going about the rotation clockwise, then 've will go counterclockwise, despite your base system

Answer 30

counterclockwise and clockwise*

Answer 31

True, I see what you mean . . . I don't know, maybe I'm just super confused or something. I know all the facts, it's just the "show it" part that's tripping me up.

Answer 32

you can explain that more in math