Question

What is the equation r(t) for 0 ≤ t ≤ π/4?

1) r(t) = sin(t), cos(t), tan(t)
2) r(t) = sin(t), cos(t), tan(t), 0 ≤ t ≤ π/4
3) r(t) = sin(t), cos(t), tan(t), 0 ≤ t ≤ π/2
4) r(t) = sin(t), cos(t), tan(t), 0 ≤ t ≤ π

Answer 1

The correct answer is option 2, which gives the vector valued function r(t) = sin(t), cos(t), tan(t), with the domain 0 ≤ t ≤ π/4, ensuring that all components of the function are defined.

Step-by-step explanation:

The question appears to ask for the correct representation of a vector valued function r(t) within a specified domain for t. The correct function will include trigonometric functions which must all be defined within the given domain for t. Looking at the options provided, sin(t) and cos(t) are always defined, but tan(t) is not defined when t equals π/2 because it would result in a division by zero since tan(t) = sin(t)/cos(t) and cos(π/2) = 0. Therefore, the correct equation must limit t such that tan(t) is defined. This allows us to conclude that the correct equation is r(t) = sin(t), cos(t), tan(t), 0 ≤ t ≤ π/4, because for all t in this interval, sin(t), cos(t), and tan(t) are all defined. This matches option 2 in the list provided by the student. Notice that in the domain 0 ≤ t ≤ π/4, tan(t) is defined because it does not reach its asymptote at π/2.