Why do sine and cosine have different x intercept patterns?

[mr.me]

hi everyone. Not really a homework question but I'm trying to teach myself trig and I wonder

Why does a sine graph have x intercepts in multiples of pi and why does a cosine graph have intercepts pi/2

 

[BloodyFrozen]

Well sine of pi radians is 0 (x-int). Cosine of pi/2 is 0

 

[mr.me]

So since sin(pi)= 0 then it is graphed as a ($\pi$,0) intercept for every integer of pi?

If that's so then why does sin(pi)=zero?

 

[Mentallic]

So since sin(pi)= 0 then it is graphed as a ($\pi$,0) intercept for every integer of pi?

If that's so then why does sin(pi)=zero?

Have you seen the circle representation of trigonometric values?

 

[Saitama]

So since sin(pi)= 0 then it is graphed as a ($\pi$,0) intercept for every integer of pi?

If that's so then why does sin(pi)=zero?

$\pi$ is in radians and in degrees it is 180^o.
sin 180^o = sin(90+90)^o = cos 90^o = 0.

 

[hunt_mat]

You're going to to come across the definition of $\pi$ sooner or later, so here is is: $\pi$ is the smallest positive number such that:
$$\cos\left(\frac{\pi}{2}\right) =0$$

 

[HallsofIvy]

$$cos(x)= - sin(x- \pi/2)$$

 

[phinds]

hi everyone. Not really a homework question but I'm trying to teach myself trig and I wonder

Why does a sine graph have x intercepts in multiples of pi and why does a cosine graph have intercepts pi/2

If you really want to learn trig, you need to study the various ways that the sine wave can be generated. Once you do that, it's (1) obvious what the answer to your question is and (2) easier to understand trig in general.

 

[phinds]

You're going to to come across the definition of $\pi$ sooner or later, so here is is: $\pi$ is the smallest positive number such that:
$$\cos\left(\frac{\pi}{2}\right) =0$$

Huh? I though the definition of pi was the ratio of the circumference of a circle to its diameter. Trig has nothing to do with it.

 

[HallsofIvy]

There generally are many different ways to define a specifice thing.

 

[phinds]

There generally are many different ways to define a specifice thing.

Yes, certainly, but I had the impression that the ratio definition of pi was made before anyone had ever invented trig and that it is in some sense a "fundamental" definition and that while there may be others that happen to be factually correct, they are unnecessary and happened after the fact.

 

[arildno]

Yes, certainly, but I had the impression that the ratio definition of pi was made before anyone had ever invented trig and that it is in some sense a "fundamental" definition and that while there may be others that happen to be factually correct, they are unnecessary and happened after the fact.

Why do you believe in some sort of historical evaluative primacy of definitions??

There are very good reasons, for example, why a base definition of sine&cosine in terms of the solutions of a specific eigenvalue problem is more interesting than the historically first definition of them.

 

[phinds]

Why do you believe in some sort of historical evaluative primacy of definitions??

Actually, I don't as a rule, especially since sometimes a better understanding of a phenomenon leads to a better definition, but somehow in this particular case it just seems like a more fundamenal definition that cannot be bettered.

There are very good reasons, for example, why a base definition of sine&cosine in terms of the solutions of a specific eigenvalue problem is more interesting than the historically first definition of them.

I can't agrue with that because my math is gone (not that I would necessarily want to argue with it anyway) but I don't understand what that has to do with the ratio definition of pi, especially in light of the fact that you are talking about trig functions and I don't see how they are needed or helpful in defining pi. VALID, I can see, but better, I cannot.

 

[hunt_mat]

Huh? I though the definition of pi was the ratio of the circumference of a circle to its diameter. Trig has nothing to do with it.

I learned the definition in my analysis course at university.