How would I calculate something like sin(1) without a calculator?

Question

zepdrix · Answer

You wouldn't.
$\Large\rm \sin(1)$ is as simple as it's going to get.

If you had been given something like $\Large\rm \sin(70^o)$ or $\Large\rm \sin(15^o)$ then you apply some tricks to relate them to your special angles 30, 45 and such.

But I don't think we can do anything with sin(1).
Maybe there are some fancy tricks that I'm not aware of though :P

jdoe0001 · Answer

did you mean $\bf sin^{-1}(1)\quad ?$

anonymous · Answer

no, what Im working on is autonomous differential equations and isoclines. So, Im given the differential equation y'=sin(y) and Im supposed to graph a small directional field for it. Now, my book tells me that isoclines for autonomous differential equations are all horizontal. So I know that along each horizontal line, there will be the same slope for each t value.

anonymous · Answer

The part Im struggling with is figuring out the slope. So, if I understand correctly, I should just take y'=sin(y) , and for each y, say, 2,  the slope should be sin(2). But, I have no idea how to figure out what that looks like without a calculator.

marissalovescats · Answer

My advice is: 

Memorize the unit circle

zepdrix · Answer

Hmm I'm not sure why you're setting y' and y to be the same values.
Your y' ( the slope of the function on the right ) can never be anything except 0, yah?
That's what they're telling you in the book.
y'=0 everywhere because the slope is horizontal.

So I think you want to do something more like this:$$\Large\rm 0=\sin(y)$$And then you would plug 0 in for your `t`'s on the right as well,
is that how the process works?
I haven't done this in a while.
Anyway, you have `t`'s on the right, so you get that the whole region is producing horizontal isoclines ....... or something...
is this helpful maybe?
blahhhhhhhh

zepdrix · Answer

you have NO* `t`'s on the right*
sorry typo

anonymous · Answer

Hehe, you've given me some things to think about, I appreciate it.