Sketch the vector function
*f*(*t*) =t, sin *t*>
for

- all real values of
*t*

Answer

- Make a table, using values of
*t* for which we can easily find the sine and cosine:

%%%%%%DATA ENTRY This data is in a table.

*t* & cost & sin t\hline0&1&0$\frac{π}{6}$&$\frac{\sqrt 3}{2}$&$\frac{1}{2}$$\frac{π}{4}$&$\frac{\sqrt 2}{2}$&$\frac{\sqrt2}{2}$$\frac{π}{3}$&$\frac{1}{2}$&$\frac{\sqrt3}{2}$$\frac{π}{2}$&0&1

0 ≤ t ≤ \frac{π}{2}These outline a quarter of a circle, which is what we find when we connect the dots:

PICTURE graph dots and the curve through themIf we replace *t* with θ, the coordinates (cos θ, sin θ) describe a point on the unit circle.Since $0 ≤ t ≤ \frac{π}{2}$, we get the portion of the unit circle where the angle goes from 0 to $\frac{π}{2}$.

- Since the coordinates are still describing the unit circle, this time we go around the whole unit circle:PICTURE: draw unit circle, label the point (1,0) with both t = 0 and t = 2π

- The coordinates are still describing the unit circle. The picture looks the same as the graph in part (B):PICTURE: vector funcs 6

However, since we're allowing ``all real values of <em>t" it's like we're traveling around the circle over and over. Every time <em>tem> is an integer multiple of 2π we pass through the point
(1,0). We'll talk more about this when we get to parametric equations.