Question Video: Converting Coordinates into Polar Coordinates | Nagwa Question Video: Converting Coordinates into Polar Coordinates | Nagwa

Question Video: Converting Coordinates into Polar Coordinates Mathematics • Third Year of Secondary School

Convert (−2, 5) to polar coordinates. Give the angle in radians and round to two decimal places.

03:22

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.

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