Problem 39 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ \left(1, \frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

(a) \( \(1, -\frac{3\pi}{2}\) \), (b) \( \(-1, \frac{3\pi}{2}\) \), (c) \( \(1, \frac{5\pi}{2}\) \).

Step 1: Plot the Given Point

Plot the point with polar coordinates \( \left(1, \frac{\pi}{2}\right) \). This means a radius \( r = 1 \) and an angle \( \theta = \frac{\pi}{2} \). This point is located 1 unit above the origin along the positive y-axis.

Step 2: Find (a) \( r > 0, \ -2\pi \leq \theta < 0 \)

For \( r = 1 \) and \( -2\pi \leq \theta < 0 \, subtract \( 2\pi \) from \theta = \frac{\pi}{2} \). Calculate: \[ \theta = \frac{\pi}{2} - 2\pi = \frac{\pi}{2} - \frac{4\pi}{2} = -\frac{3\pi}{2} \] Thus, the coordinates are \( \(1, -\frac{3\pi}{2}\) \).

Step 3: Find (b) \( r < 0, \ 0 \leq \theta < 2\pi \)

For \( r = -1 \) and \( 0 \leq \theta < 2\pi \, add \pi \to \theta = \frac{\pi}{2} \). Compute: \[ \theta = \frac{\pi}{2} + \pi = \frac{\pi}{2} + \frac{2\pi}{2} = \frac{3\pi}{2} \] Thus, the coordinates are \( \(-1, \frac{3\pi}{2}\) \).

Step 4: Find (c) \( r > 0, \ 2\pi \leq \theta < 4\pi \)

For \( r = 1 \) and \( 2\pi \leq \theta < 4\pi \, add \2\pi \to \theta = \frac{\pi}{2} \). Calculate: \[ \theta = \frac{\pi}{2} + 2\pi = \frac{\pi}{2} + \frac{4\pi}{2} = \frac{5\pi}{2} \] Thus, the coordinates are \( \(1, \frac{5\pi}{2}\) \).

Key Concepts

plotting polar coordinates

Plotting points in polar coordinates is different from plotting in Cartesian coordinates. Instead of using x and y values, polar coordinates use a radius (r) and an angle (θ). The radius tells you how far the point is from the origin, and the angle tells you the direction from the origin. In the given problem, we plot the point \((1, \frac{\pi}{2})\). This means the radius is 1 unit, and the angle is \(\frac{\pi}{2}\), which corresponds to 90 degrees. So, the point is located 1 unit above the origin along the positive y-axis.

converting angles

Angles in polar coordinates can be tricky because they can have multiple representations. For instance, adding or subtracting \(2\pi\) (or 360 degrees) to an angle will land you on the same direction. In the exercise, we convert angles to fit within the required ranges. To place an angle between \(-2\pi\) and 0, we subtract \(2\pi\) from \( \frac{\pi}{2} \), resulting in \( -\frac{3\pi}{2} \). For positioning between 0 and \(2\pi\), we add \(\pi\) to \( \frac{\pi}{2} \), resulting in \( \frac{3\pi}{2} \). Lastly, to fit an angle between \(2\pi\) and \(4\pi\), we add \(2\pi\) to it, resulting in \( \frac{5\pi}{2} \).

coordinate transformation

Coordinate transformation involves changing a point from one coordinate system to another or modifying a point's existing coordinates to fit specific criteria. In this exercise, we transform the given polar coordinates to fit different requirements by adjusting the radius and angle. For instance:

• When \( r > 0 \) and \( -2\pi \leq \theta < 0 \), we subtract \(2\pi\) from the angle \( \frac{\pi}{2} \).
• When \( r < 0 \) and \( 0 \leq \theta < 2\pi \), we add \(\pi\) to the angle.
• When \( r > 0 \) and \( 2\pi \leq \theta < 4\pi \), we add \(2\pi\) to the angle.

These transformations allow us to work within the required range of angles while keeping the point's position consistent.

trigonometry

Trigonometry is deeply embedded in polar coordinates because it deals with angles and distances. Angles are measured in radians or degrees, and functions like sine and cosine are used to understand the position of a point with respect to the origin. For example, in converting angles:

• We know that subtracting \( 2\pi \) rotates our angle but keeps our point's position.
• Adding \( \pi \) and adjusting the radius with -1 effectively flips the point, but keeps it in the same direction.
• Adding \( 2\pi \) rotates the angle full circle and moves into the next cycle.

These concepts are fundamental in understanding how to work with polar coordinates and their transformations. Mastering these allows for better manipulation and plotting of points in the polar system.