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.

tan(α)　~ sin(α) ~ α = (pi*α')/(180 deg)

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

The absolute error is: \[e _{abs} = \left| \tan (\alpha) - \alpha \right|\]
The percent error is : \[e _{} = \left| \tan (\alpha) - \alpha \right| \times 100 / \tan (\alpha) \]
for alpha = 0.53 ; e = 9.56
for alpha = 0.55 : e = 10.33
The approximation is good until alpha = 0.53 radians is equal 30 degrees
Attached plot of y=tan(alpha) and y=alpha