Video Transcript
Convert negative two, five to polar
coordinates. Give the angle in radians and round
to two decimal places.
In this question, we need to
convert Cartesian coordinates to polar coordinates, where Cartesian coordinates are
written in the form 𝑥, 𝑦 and polar coordinates, in the form 𝑟, 𝜃.
There are some general formulae
that we can use to evaluate these. 𝑟 is equal to the square root of
𝑥 squared plus 𝑦 squared, and 𝜃 is equal to the inverse tan of 𝑦 over 𝑥. However, we do need to be very
careful when calculating our value of 𝜃. To understand why, let’s consider
the two-dimensional 𝑥𝑦-coordinate plane.
The angle 𝜃 is measured in the
counterclockwise direction from the positive 𝑥-axis. We can therefore mark on the angles
in radians as shown. The point in this question has
Cartesian coordinates negative two, five. This means that it lies in the
second quadrant and the angle 𝜃 is as shown. The value of 𝑟 is the magnitude or
length of the line from the origin to the point negative two, five.
We can calculate this value using
the Pythagorean theorem in the right triangle. The two shorter sides have lengths
two units and five units. So 𝑟 squared is equal to two
squared plus five squared. Taking the square root of both
sides and since 𝑟 must be positive, we have 𝑟 is equal to the square root of four
plus 25. This is equal to the square root of
29, which is equal to 5.3851 and so on. And to two decimal places, 𝑟 is
equal to 5.39.
We can calculate the measure of
angle 𝛼 in our right triangle using our knowledge of the trigonometric ratios. As the tan of any angle 𝛼 is equal
to the opposite over the adjacent, we have tan 𝛼 is equal to five over two. We can then take the inverse
tangent of both sides. This gives us 𝛼 is equal to 1.1902
and so on. Since the angles 𝜃 and 𝛼 sum to
𝜋 radians, 𝜃 is equal to 𝜋 minus 𝛼. And this is equal to 1.9513 and so
on. Once again, rounding to two decimal
places, we have 𝜃 is equal to 1.95.
The Cartesian coordinates negative
two, five written in polar form to two decimal places are 5.39, 1.95.
