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In physics it is important to use mathematical approximations. Demonstrate that for small angles (< 20°) the following relationship is true where α is in radians and α' is in degrees.


Use a calculator to find the largest angle for which tan α may be approximated by α with an error less than 10.0%.

Verified Answer :

meerkat18

we are required in this problem to approximate tan α which α is less than 20 degrees.
In radians 20degrees is equal to 0.349; find tan 0.349=0.363
 percentage error=(0.349-.363)*100/(0.363) =-3 percent error so for largest angle 20 degrees , the error is less than 10 percent

Answer:

\(\alpha = 30.9 degree\)

Explanation:

percentage error of approximation is less than 10%

so we will have

\(error = \frac{tan\alpha - \alpha}{tan\alpha} \times 100\)

now we will have

\(10 = \frac{tan\alpha - \alpha}{tan\alpha} \times 100\)

\(0.1 = \frac{tan\alpha - \alpha}{tan\alpha}\)

now we have

\(0.1 tan\alpha = tan\alpha - \alpha\)

\(0.9 tan\aplha = \alpha\)

so here we have

\(\alpha = 30.9 degree\)

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