Tangent line help [10 Points!!]

An equation of the line tangent to the graph of y = cos(2x) at x = pi/4?

I know the derivative of cos(2x) is -2sin(2x). Where do i go from there? What do I plug in where? A step by step would be helpful

Thanks.

Okay, so what you've found is the derivative of the line at a given point, (x,y). Because the derivative is in terms of x, you only need the x value of that point, in this case, x= pi/4. The question is asking you to find the line that has the same slope as the derivative at x=pi/4, so what you have to do is plug pi/4 into the derivative:
-2sin(2pi/4)=-2(1)= -2
now you have the slope of the tangent line. To write the equation of a line, you need a point. This point is on the original function, the point of tangency. You already have the x value of the point pi/4, so now you plug pi/4 into the original equation to find the y value of that point:
y=cos(2pi/4)=cos(pi/2)=0.
Now you know that when x=pi/4 at the original function, y=0. Now, all you have to do is plug the x value, y value, and slope into the point-slope form of a line equation and, at least according to the AP board, you're done. Your individual teacher may require more simplification.
point slope: y-0=1(x-(pi/4)).
A cleaner version: y=x-(pi/4)
I hope this helps!

what is the actual question? if you're looking for the equation of the tangent, the connection between the tangent to a curve is that their slopes are equal at the point where they meet.

you know that the slope of the curve is -2sin(2x) as the slope is the derivation of the equation of the curve however this is the general slope of the curve.

x = pi/4 at the point where the curve and the tangent meet, so sub in pi/4 for x in the slope to find the actual slope at that point.

slope = -2sin(2x)
= -2sin(2(pi/4))
= -2sin(pi/2)
=-2(1)
= -2

the slope of your tangent then is -2

to get the equation of a line you need the slope and a point on the line.

you can find a point on the tangent by finding the y co ordinate of the point where the tangent and the curve meet as you already have the x co ordinate

in order to do this you can sub x = pi/4 into the equation of the curve: y = cos(2x)

y = cos(2(pi/4))
y = cos(pi/2)
y= 0

so your point is (pi/4 , 0)

now you can get your equation of the tangent with slope: -2 and point: (pi/4 , 0)

formula for equation of a line: y-y1=m(x-x1)

y-0 = -2(x-pi/4)
y = -2x+pi/2
2x + y = pi/2

Hope that helps!

equation
y - f(pi/4) = f '(pi/4) (x - (pi/4))