Question

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%.

Answer 1

`\alpha = 30.9 degree`

Step-by-step explanation:

percentage error of approximation is less than 10%

so we will have

`error = (tan\alpha - \alpha)/(tan\alpha) * 100`

now we will have

`10 = (tan\alpha - \alpha)/(tan\alpha) * 100`

`0.1 = (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`

Answer 2

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