Question

In physics, it is important to use mathematical approximations. for instance, in a small angle approximation, tanα ∼ sin α . find the largest angle α for which the difference between the sine and the tangent of an angle is less than 10.4% of the sinα value. answer in units of rad.

Answer 1

Note that x is measured in radians.
The percent difference or error between sin(x) and tan(x) is calculated as 
$d = 100( \frac{|sin(x)-tan(x)|}{sin(x)} )$

We want this percent difference to be less than 10.4%.
Because tan(x) > sin(x) for small values of x, define
$f(x) = \frac{tan(x)-sin(x)}{sin(x)}= 0.104 \\ or \\ sec(x) - 1 = 0.104 \\ sec(x) = 1.104$

From the calculator, obtain
sec⁻¹ 1.104 = 0.4375

Answer:  x = 0.4375 radians