# Video: Evaluating Vector-Valued Functions

For the given function π(π‘) = 2 csc (2π‘) π + 3 tan (π‘) π, evaluate π(π/4).

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### Video Transcript

For the given function π of π‘ equals two csc two π‘ π plus three tan π‘ π, evaluate π of π by four.

Here, π is a vector-valued function. Itβs a function whose range is a set of vectors. And we can evaluate π of π by four in much the usual way by substituting π‘ equals π by four into each component function. For the horizontal component, the component of π, we get two csc two times π by four. Of course, we know that csc π₯ is equal to one over sin π₯ 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 π. Thatβs three tan of π by four. And of course, tan of π by four is simply one. So thatβs three times one, which is three. That means that π of π by four is two π plus three π. Contextually, this gives us the position of a point in the plane when π‘ is equal to π by four.