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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%.

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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:

[tex]\alpha = 30.9 degree[/tex]

Explanation:

percentage error of approximation is less than 10%

so we will have

[tex]error = \frac{tan\alpha - \alpha}{tan\alpha} \times 100[/tex]

now we will have

[tex]10 = \frac{tan\alpha - \alpha}{tan\alpha} \times 100[/tex]

[tex]0.1 = \frac{tan\alpha - \alpha}{tan\alpha}[/tex]

now we have

[tex]0.1 tan\alpha = tan\alpha - \alpha[/tex]

[tex]0.9 tan\aplha = \alpha[/tex]

so here we have

[tex]\alpha = 30.9 degree[/tex]

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