Determining coordinates via Projection

The x- and y-axis are the horizontal and vertical lines intersecting at the circle's centre.

Figure 8: The vector coordinates are determined by projecting onto the x and y-axis.
$\begin{figure}\centerline{\scalebox{0.7}{% \input{circ_xyaxis.pstex_t}}}\end{figure}$
Projecting¹ the vector onto the x- and y-axes allows us to determine its coordinates in the xy-plane.
If the vector is rotated in a counterclockwise direction, at angle $\theta$ from the positive x-axis, projecting onto both the x- and y-axes creates right angle triangles.
Trigonometric identities, with knowledge of $\theta$ and the vector length, will help us determine the coordinates:

$\displaystyle y = r\sin(\theta) \qquad x = r\cos(\theta).$

**Figure 8:** The vector coordinates are determined by projecting onto the x and y-axis.
$\begin{figure}\centerline{\scalebox{0.7}{% \input{circ_xyaxis.pstex_t}}}\end{figure}$

``Music 270a: Sinusoids'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).