Why do some trigonometric functions need to be worked out to a higher degree than others using power series?

Question

Using power series for $\sin{x}$ why do for some values of $x$ it needs to be worked out more than others? $$\sin{x}=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^7}{7!}+...$$ From this lets say for the value of $\sin 1$ radian , it takes only 2 terms to get an approximate enough answer of $0.8333..$ But for $\sin 5$ radians it takes 7 terms to get an approximate answer. Why is it so? Till which terms would be plausible to use in my exams if I really needed to calculate sin values(they don't allow calculators)