For the given function 𝑟 of 𝑡
equals two csc two 𝑡 𝑖 plus three tan 𝑡 𝑗, evaluate 𝑟 of 𝜋 by four.
Here, 𝑟 is a vector-valued
function. It’s a function whose range is a
set of vectors. And we can evaluate 𝑟 of 𝜋 by
four in much the usual way by substituting 𝑡 equals 𝜋 by four into each component
function. For the horizontal component, the
component of 𝑖, we get two csc two times 𝜋 by four. Of course, we know that csc 𝑥 is
equal to one over sin 𝑥 and two times 𝜋 by four is equal to 𝜋 over two. So this becomes two over sin of 𝜋
by two. And since sin 𝜋 by two is simply
one, we end up with two divided by one, which is two. We repeat this for the vertical
component for 𝑗. That’s three tan of 𝜋 by four. And of course, tan of 𝜋 by four is
simply one. So that’s three times one, which is
three. That means that 𝑟 of 𝜋 by four is
two 𝑖 plus three 𝑗. Contextually, this gives us the
position of a point in the plane when 𝑡 is equal to 𝜋 by four.