Finding Area Between Cosine Curves [0, pi] Using Integration

[char808]

Homework Statement

Find the area between the curves y=4cosx and y = 4cos(2x) [0, pi]

Homework Equations

The Attempt at a Solution

I know I need to integrate this, but I get hung up finding the intersection of the two lines so I can split it into two different areas.

4cosx = 4 cos(2x)

0 = cos(2x) - cos(x) I think I need to use a trig identity here, but I'm not sure.
0 = sin(2x)sin(x)

x= pi?

 

[Mark44]

Homework Statement

Find the area between the curves y=4cosx and y = 4cos(2x) [0, pi]

Homework Equations

The Attempt at a Solution

I know I need to integrate this, but I get hung up finding the intersection of the two lines so I can split it into two different areas.

4cosx = 4 cos(2x)

0 = cos(2x) - cos(x) I think I need to use a trig identity here, but I'm not sure.

Finding the points of intersection would be a very good idea, and a trig identity would be very useful.

0 = sin(2x)sin(x)

?? How did you go from cos(2x) - cos(x) to sin(2x)sin(x)?

x= pi?

That looks like a guess.

 

[char808]

Yes, I'm at a loss about where to go from here:

0 = cos(2x) - cos(x)

Possibility: (I am admittedly weak when it comes to trig, I could/can figure this out with almost any other function.)

0=cos(2x-x)
0= cos(x)

x= pi/2?

 

[Char. Limit]

0 = cos(2x) - cos(x) I think I need to use a trig identity here, but I'm not sure.

Yes indeed you do. What identities do you know for cos(2x)?

 

[char808]

I don't. If you have a reference link I can go study I would appreciate that. Not covered in the book I have.

 

[Mark44]

Yes, I'm at a loss about where to go from here:

0 = cos(2x) - cos(x)

Possibility: (I am admittedly weak when it comes to trig, I could/can figure this out with almost any other function.)

0=cos(2x-x)

cos(2x) - cos(x) $\neq$ cos(2x - x)

0= cos(x)

x= pi/2


?

 

[Mark44]

I don't. If you have a reference link I can go study I would appreciate that. Not covered in the book I have.

Did you study trig at any time? If so, what did you do with your book?

khanacademy.org has a lot of lectures about a variety of math stuff. You might start there.

 

[Char. Limit]

Guessing also works. For example, cos(2x)=cos(x) means that, at one point, 2x=x (there are also other points, however). Only one point can satisfy 2x=x.

 

[char808]

Did you study trig at any time? If so, what did you do with your book?

khanacademy.org has a lot of lectures about a variety of math stuff. You might start there.

I last studied trig ~7-8 years ago. I probably sold the book.

I will check out the link.

 

[tiny-tim]

hi char808 !

you need to learn the standard trigonometric identities …

cosA - cosB = 2 sin((A+B)/2) sin((A-B)/2) would help

 

[Mark44]

One that I found helpful was
cos(2x) = cos²(x) - sin²(x)

Two other forms of this are
cos(2x) = 2cos²(x) - 1
cos(2x) = 1 - 2sin²(x)

One of these can be used to write your equation cos(2x) = cos(x) as a quadratic in form.