# Using differentials to estimate error

• MHB
• Buka
In summary, the conversation discussed using differentials to estimate the error in computing the length of the hypotenuse of a right triangle with one side known to be 12 cm and the opposite angle measured as 30° with a possible error of ±1°. The process involved using the formula c=12(sin(θ))^-1 and differentiating to find dc= -12(sin(θ))^-2 cos(θ) dθ. The conversation also emphasized the importance of showing progress and providing thoughts when asking for help.
Buka
one side of a right triangle is known to be 12 cm long and the opposite angle is measured as 30°, with a possible error of ±1°.
(a) Use differentials to estimate the error in computing the length of the hypotenuse.

Hello and welcome to MHB! :D

I have retitled your thread so that the title now reflects the nature of the question being asked.

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?

Buka said:
one side of a right triangle is known to be 12 cm long and the opposite angle is measured as 30°, with a possible error of ±1°.
(a) Use differentials to estimate the error in computing the length of the hypotenuse.
$$\displaystyle sin(\theta)= \frac{a}{c}$$ so that $$\displaystyle c= \frac{a}{sin(\theta)}= \frac{12}{sin(\theta)}= 12(sin(\theta))^{-1}$$. Differentiating, $$\displaystyle dc= -12(sin(\theta))^{-2} cos(\theta) d\theta$$.

Take $$\displaystyle \theta= 30^o$$ and $$\displaystyle d\theta= \pm\frac{\pi}{180}(1)= \pm\frac{\pi}{180}$$ (because the derivative of sine assumes the angle is in radians, not degrees).

## What is the purpose of using differentials to estimate error?

The purpose of using differentials to estimate error is to find an approximation for the error in a given measurement or calculation. This can be useful in various scientific fields, such as physics, engineering, and statistics.

## How does the differential method work?

The differential method involves using the derivative of a function to estimate the change in the function's output when the input changes by a small amount. This change can then be used to approximate the error in the function's output.

## What are the advantages of using differentials to estimate error?

One advantage of using differentials is that it allows for a quick and easy way to estimate error without having to perform complex calculations. It also provides a more accurate approximation compared to other methods, such as linearization.

## What types of errors can be estimated using differentials?

Any type of error that can be expressed as a change in a function's output due to a change in its input can be estimated using differentials. This includes errors in measurements, calculations, and predictions.

## What are some potential limitations of using differentials to estimate error?

One limitation is that it assumes a linear relationship between the input and output of a function, which may not always be the case. It also relies on small changes in the input, so it may not be accurate for larger changes. Additionally, it may not account for all sources of error in a given measurement or calculation.

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