Final answer:
To replace sin x by x − x³/6 with an error of magnitude no greater than 6×10⁻⁴, the values of x should be approximately less than or equal to 0.4.
Step-by-step explanation:
To replace sin x by x − x3/6 with an error of magnitude no greater than 6×10−4, we need to determine the values of x for which the difference between sin x and x − x3/6 is less than or equal to 6×10−4. Let's consider the Taylor series expansion of sin x: sin x = x − x3/6 + x5/120 + ...
By comparing the terms, we can see that the error term x5/120 is smaller than 6×10−4 when x is around 0.4 or smaller. Therefore, we can replace sin x by x − x3/6 with an error of magnitude no greater than 6×10−4 for values of x approximately less than or equal to 0.4.