\(\displaystyle sin(\theta)= \frac{a}{c}\) so that \(\displaystyle c= \frac{a}{sin(\theta)}= \frac{12}{sin(\theta)}= 12(sin(\theta))^{-1}\). Differentiating, \(\displaystyle dc= -12(sin(\theta))^{-2} cos(\theta) d\theta\).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.
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.
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.
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.
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.
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.