Triangle Approximation Derivation

• GroupTheory1
In summary, the conversation discusses a drawing with variables and the use of the law of cosines to derive an approximation. The approximation is then discussed and compared to the original equation, with the mention of a possible use of Taylor series to achieve the given form.

Homework Statement

Here is a drawing with all the needed variables:

http://i.imgur.com/192GI.jpg

The Attempt at a Solution

I have been trying to figure out how this approximation is derived for some time now and have no progress to show for it. Any help in figuring out the steps would be greatly appreciated.

The law of cosines says that

b^2 - a^2 = c^2 - 2ac * cos(B),

so,

b - a = -2ac * cos(B) / (b+a) + c^2 / (b+a).

If one says that a ~ b because c << a, then the above becomes

b - a ~ -c * cos(B) + c^2 / (2a),

which is close but does not match and is way off for B approaching zero degrees.

GroupTheory1 said:
The law of cosines says that

b^2 - a^2 = c^2 - 2ac * cos(B),

so,

b - a = -2ac * cos(B) / (b+a) + c^2 / (b+a).

If one says that a ~ b because c << a, then the above becomes

b - a ~ -c * cos(B) + c^2 / (2a),

which is close but does not match and is way off for B approaching zero degrees.

As I read the figure and understand your notation, B = $$\pi$$ - ($$\theta$$ + $$\gamma$$), thus what you got equals what's on the figure.

Sourabh N said:
As I read the figure and understand your notation, B = $$\pi$$ - ($$\theta$$ + $$\gamma$$), thus what you got equals what's on the figure.

Almost but not quite. I cannot figure out where the sin^2(B) factor comes from.

Are you familiar with Taylor series? I could get the form they have given using a Taylor series approximation.

What is the Triangle Approximation Derivation method?

The Triangle Approximation Derivation method is a mathematical technique used to estimate the area under a curve by dividing the curve into smaller triangles and calculating the area of each triangle. The sum of all the triangle areas gives an approximation of the total area under the curve.

How is the Triangle Approximation Derivation method used in science?

The Triangle Approximation Derivation method is commonly used in science to calculate the area under a curve in situations where the exact area cannot be easily determined. This method is particularly useful in physics, chemistry, and biology to estimate the volume of irregularly shaped objects or the amount of a substance in a solution.

What are the limitations of the Triangle Approximation Derivation method?

The Triangle Approximation Derivation method is an approximation technique and therefore may not give an exact value for the area under a curve. The accuracy of the method depends on the size of the triangles used, with smaller triangles giving a more accurate result. Additionally, this method may not be suitable for curves with complex shapes or sharp turns.

How can the accuracy of the Triangle Approximation Derivation method be improved?

The accuracy of the Triangle Approximation Derivation method can be improved by using smaller triangles and increasing the number of triangles used. Additionally, using a computer or graphing calculator to calculate the area of each triangle can further improve accuracy. It is also important to carefully choose the starting and ending points of the curve to minimize errors.

Are there any real-life applications of the Triangle Approximation Derivation method?

Yes, the Triangle Approximation Derivation method has many real-life applications, such as estimating the area of irregularly shaped fields for agricultural purposes, calculating the volume of a lake or reservoir, and determining the amount of medication in a patient's blood over time. This method is also used in engineering and construction to estimate the amount of materials needed for a project.

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