A First Course in Linear Algebra, 2017a

8.1. Polar Coordinates and Polar Graphs 435
Solution. For the point specified by the polar coordinates (−5,π/6), r = −5, and xθ = π/6. From 8.1
( π
)
x = r cos(θ)=−5cos = − 5 3
6 2√
( π
)
y = r sin(θ)=−5sin = − 5 6 2
Thus the Cartesian coordinates are ( − 5 2√
3,−
5
2
)
. The point is shown in the following graph.
(− 5 2√
3,−
5
2
)
Recall from the previous example that for the point specified by (5,π/6), the Cartesian coordinates
are ( √ )
5
2
3,
5
2
. Notice that in this example, by multiplying r by −1, the resulting Cartesian coordinates are
also multiplied by −1.
♠
The following picture exhibits both points in the above two examples to emphasize how they are just
on opposite sides of (0,0) but at the same distance from (0,0).
( 5 2√
3,
5
2
)
(− 5 2√
3,−
5
2
)
In the next two examples, we look at how to convert Cartesian coordinates to polar coordinates.
Example 8.3: Finding Polar Coordinates
Suppose the Cartesian coordinates of a point are (3,4). Find a pair of polar coordinates which
correspond to this point.
Solution. Using equation 8.2, we can find r and θ. Hence r = √ 3 2 + 4 2 = 5. It remains to identify the
angle θ between the positive x axis and the line from the origin to the point. Since both the x and y values
are positive, the point is in the first quadrant. Therefore, θ is between 0 and π/2 . Using this and 8.2, we
have to solve:
tan(θ)= 4 3