# Slope of sine

- Author:
- Allen Olson

- Topic:
- Sine

1. Start by dragging the point

*X*to the right and to the left. You should find that it stays on the function*f*(*t*) plotted in gold. This function is sin(*t*), so*X*is showing the value of the sine function at different times.2. GeoGebra can determine the line that is tangent to a curve at any point. To see that tangent line for

*X*, click on the circle next to "a:" in the "Line" section on the left side of the display above. Then you should be able to see what happens as you drag*X*to the left and to the right. 3. The slope of the tangent line is taken to be the instantaneous slope of the curve, and the point "*V*" has been set equal to that slope. To see*V*, click on the circle next to "V =" in the "Point" section on the left side of the display. You should see, as you drag*X*back and forth, that the value of*V*changes and that it matches the slope of the line*a*. You may want to turn off line*a*at this point, by clicking on the circle next to it on the left side of the display. 4. We can think of*V*as itself tracing out some function, and in fact we can see that function by showing*g*(*t*) by clicking on the circle next to "g(t) =" in the "Function" section on the left side of the display. What function is this? It's the cosine function! So the slope (or derivative) of sine is cosine. 5. We can repeat this process with the function*g*(*t*). That is, we can find the slope of it at any point. We just have GeoGebra draw a tangent line to the curve where V is. This line is*b*if you want to show it; and the value of the slope of that tangent line is the point*A*, which you can also make visible.6. What function does the point *A* trace out? To get a hint, turn on the function *h*(*t*). How does *h*(*t*) (in blue) compare to *f*(*t*) (in gold)?

If we think of *X* as marking the position of an object at some time *t*, then *V* represents the instantaneous change in position as a function of time which is the velocity. What does *A* represent? (Hint: It starts with an 'a'.)