Simplifying a arc length problem

In summary, the conversation discusses simplifying an arc length problem using Maple software. The problem involves finding the arc length of L= Int(-2..2) sqrt(16*cosh(4*t)^2+9*sinh(4*t)^2+9). After using Maple to simplify the problem to sqrt(25*cosh(4*t)^2), the individual is unsure of the steps taken and requests assistance. Through the conversation, they are able to apply the hyperbolic equivalent of sin^2 + cos^2 = 1 and simplify the problem to 9 + 9*sinh^2 (4t). They thank the other person for the hint and their help in solving the problem.
  • #1
hutchwilco
4
0
Simplifying an arc length problem

I have L= Int(-2..2) sqrt(16*cosh(4*t)^2+9*sinh(4*t)^2+9)

and can use Maple to simplify this to sqrt(25*cosh(4*t)^2)
but I just can't see how that is done. (or how to get maple to show me the steps!)

Can anyone help by showing the steps, including any trig substitutions/identities used?
 
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  • #2
try the definition of cosh and sinh in terms of exponentials (or use that to derive the hyperbolic equivalent of sin^2 + cos^2 = 1)
 
  • #3
[tex] 1+ \sinh^2 x = ...[/tex] so [tex] 9 + 9 \sinh^2 (4t) = .....[/tex]?
 
  • #4
ah! thanks - i can't believe I didn't see that - it's only one step! I often have problems seeing the stripped down identities and applying them to problems which have other factors (eg 9 and 4t etc).
Thanks for the hint
 

1. What is an arc length problem?

An arc length problem involves finding the length of a curved line segment, known as an arc, on a circle or other curved shape. This length is typically measured in units such as inches, centimeters, or degrees.

2. How is an arc length problem simplified?

To simplify an arc length problem, you can use the formula L = rθ, where L is the arc length, r is the radius of the circle, and θ is the central angle subtended by the arc. You can also use trigonometric functions such as sine and cosine to find the length of an arc.

3. When is it necessary to simplify an arc length problem?

Simplifying an arc length problem is necessary when the given information is not in a form that can be directly used in the formula L = rθ. For example, if the central angle is given in radians, you may need to convert it to degrees before using the formula.

4. What are some common challenges in simplifying an arc length problem?

Some common challenges in simplifying an arc length problem include ensuring that all units are consistent, converting between radians and degrees, and using the correct formula for the given problem. It is also important to pay attention to any given restrictions or assumptions, such as only finding the arc length in a specific quadrant of a circle.

5. Can simplifying an arc length problem be applied to other shapes besides circles?

Yes, the concept of finding the length of an arc can be applied to other curved shapes, such as ellipses or parabolas. The formula used may differ depending on the shape, but the general process of simplifying the problem remains the same.

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