Question

i learned cosine and sine definitions but i never know how to justify them why they are like that.
@Kainui

Answer 1

Okay kainui ^_^ I appreciate any help to get this

Answer 2

Have you ever defined cosine and sine using a unit circle?

Answer 3

don't want that definition-_-
we have to assume many things considering the unit circle!
i'm not looking for that

Answer 4

Actually the unit circle definition of cosine and since assume you know
the triangle definition, so that's no a good point to start at lol

Answer 5

Then what are you asking here?

Answer 6

i'm looking for justification of why sine and cosine are defined to be what they are hehe

Answer 7

So you want to know why sine is opposite over hypotenuse and why cosine is adjacent over hypotenuse?

Answer 8

yes!

Answer 9

I'll get straight to the point.
That's the same thing as asking "why is 1 a natural number"

Answer 10

That's because definitions are created, not justified.
For example, if the mathematicians wanted, they could've switched the cosine and sine.
We can't do anything about it because we're not the ones who defined trig.

Answer 11

All mathematicians agreed to the definition of the trig, so we're learning it, too.

Answer 12

No that's not a valid answer!
definitions are derived after observations and intuition
it is not like a mathematician just came up and said cos is this and sine is that
mathematically what you said is invalid

Answer 13

mathematician agreed to the rule because they understood the thought behind it!

Answer 14

i meant mathematicians

Answer 15

Actually, part of what you said was right.
To be honest, no one really knows how the ancient greeks or babylonians got their definition of trig.
My guess is that while playing around with right triangles, they (accidentally) discovered that the ratio of the sides are constant when the angle is the same.

Answer 16

And that primitive definition of trig was expanded by many prominent mathematicians such as Euler, now accepted by all of the mathematicians.

Answer 17

So, yes. I was wrong the first time.

Answer 18

well there is always explanation in math. when i said justify i meant
to understand the intuition part of this! you said no one knows, i don't think that's true
otherwise they would adapt such definition if they hadn't understood what it was?
mathematician are always look deep into what they handle they won't just adapt a rule if they haven't understood it and know the the thought of how we got to it

Answer 19

how it developed is not in my interest for now! i need to start from the bottom line of this lol

Answer 20

I'll always keep that in my mind. Thank you

Answer 21

You have my thank for sharing your thoughts about this^_^
and you are welcome!

Answer 22

I'm picking geometry to restudy it and feel that i know what this is.Because all i have done
is memorize stuff and apply them hehe

Answer 23

same.
I'm restarting from euclidean geometry. I just finished the basics of the triangle

Answer 24

@science0229 oh nice to know that!
i was asking kainui about what books could be useful to me restudying euclid geometry

Answer 25

wht definition you wanna justify ?
type it fully plz :D

Answer 26

like \(\Large \tt\color{blueviolet}{sin\Theta=opp/hyp\\
cos\Theta=adj/hyp}\) how to we approach this ratios?

Answer 27

@Kainui forgot this? lol