# Video: Evaluating Vector-Valued Functions

For the given function 𝑟(𝑡) = 2 csc (2𝑡) 𝑖 + 3 tan (𝑡) 𝑗, evaluate 𝑟(𝜋/4).

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### Video Transcript

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.