Why is the derivative of sine the cosine in radians but not in degrees? In radians the derivative of sine is the cosine. But why isn't this the same in degrees? According to this I'd first have to convert it to radians before this works. But when you look at the graph of sine whether the x is in degrees or radians, the graph stays the same... I don't understand.
Here's a graph of $y = \sin x$.  



The red line is $\sin x$ if $x$ is interpreted as an angle in degrees, and the blue line is $\sin x$ if $x$ is interpreted as an angle in radians.  See the difference?

Here's a zoomed-in version



What is $\sin(1.57\text{ rad}),$ and what is $\sin(1.57˚)?$

Intuitively, you can think of $\sin(x˚)$ as a "stretched out" version of the original graph.  As a result, the slope of the tangent flattens, changing the derivative.
this is because of the contraction of the functions as the domain changes. in degrees 0 to90 corresponds to 0 to 3.14 radians hence suitably as the general derivative is made with radians as domains we have convert other units into the general.