Question

Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. \(\left(2, \frac{7 \pi}{4}\right)\)

Short answer

Answer: The two alternative representations of the given polar coordinates are \((2, \frac{15\pi}{4})\) and \((2, -\frac{\pi}{4})\).

Step-by-step solution

Understand and convert the polar coordinates to Cartesian coordinates

The given polar coordinate is \((2, \frac{7\pi}{4})\), where the radial distance \(r = 2\) and the polar angle \(\theta = \frac{7\pi}{4}\). To convert these polar coordinates to Cartesian coordinates (x, y), we can use the following formulas: $$ x = r \cos\theta \\ y = r \sin\theta $$ Plugging in the values for \(r\) and \(\theta\): $$ x = 2\cos\left(\frac{7\pi}{4}\right) \\ y = 2\sin\left(\frac{7\pi}{4}\right) $$ Now, use the trigonometric properties of cosine and sine functions to find the x and y coordinates: $$ x = 2\left(\frac{1}{\sqrt{2}}\right) = \sqrt{2} \\ y = 2\left(-\frac{1}{\sqrt{2}}\right) = -\sqrt{2} $$ So, the Cartesian coordinates equivalent to the given polar coordinates are \((\sqrt{2}, -\sqrt{2})\).

Graph the point in Cartesian coordinates

Now that we have converted the polar coordinates to Cartesian coordinates, we can graph the point \((\sqrt{2}, -\sqrt{2})\): 1. Find the values for x and y on their respective axes. 2. Mark the point corresponding to x and y values. 3. Indicate the orientation of the angle by drawing an arrow starting from the origin, passing through the point, and moving in the anticlockwise direction.

Find two alternative representations of the polar coordinates

By adding or subtracting multiples of \(2\pi\) to the original angle, \(\frac{7\pi}{4}\), we can obtain alternative representations of the polar coordinates while keeping the radial distance (\(r\)) constant: 1. Add a full rotation to the original angle: \(\frac{7\pi}{4} + 2\pi = \frac{15\pi}{4}\), leading to an alternative representation of \((2, \frac{15\pi}{4})\). 2. Subtract a full rotation from the original angle: \(\frac{7\pi}{4} - 2\pi = -\frac{\pi}{4}\), giving another alternative representation of \((2, -\frac{\pi}{4})\). Both \((2, \frac{15\pi}{4})\) and \((2, -\frac{\pi}{4})\) would represent the same point in the Cartesian plane as \((2, \frac{7\pi}{4})\).

Key concepts

Cartesian Coordinates Conversion

Converting between polar and Cartesian coordinates can feel tricky at first, but with the right formulas, you can master it in no time. Polar coordinates, denoted as \((r, \theta)\), use a radial distance \(r\) and an angle \(\theta\) to localize a point. To switch to Cartesian coordinates, \((x, y)\), we use the following equations:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)

Plug in your values for \(r\) and \(\theta\) to get the specific coordinates. **Example:** For \((2, \frac{7\pi}{4})\), use trigonometric identities to compute

• \(x = 2 \cdot \frac{1}{\sqrt{2}} = \sqrt{2}\)
• \(y = 2 \cdot -\frac{1}{\sqrt{2}} = -\sqrt{2}\)

Thus, the Cartesian coordinates are \((\sqrt{2}, -\sqrt{2})\). This translates the point into the familiar x-y plane you often use in algebra.

Trigonometric Functions

Trigonometric functions play a vital role in the conversion of coordinates, and understanding them is key to navigating these transformations. Sine and cosine are used to determine the x and y components, respectively.These functions are based on unit circle properties:

• **Cosine** is the x-coordinate of a point on the unit circle.
• **Sine** is the y-coordinate of a point on the unit circle.

The unit circle has a radius of 1, which makes it easy to apply for angles given in radians like \(\frac{7\pi}{4}\). At this angle, you can use the known cosine and sine values:

• \(\cos\left(\frac{7\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
• \(\sin\left(\frac{7\pi}{4}\right) = -\frac{1}{\sqrt{2}}\)

These help convert our polar coordinates to Cartesian coordinates accurately. Remember, understanding these functions and their properties on the unit circle not only helps in conversions but also is fundamental in trigonometry-based problems.

Graphing Points

Graphing points in either coordinate system involves understanding how these systems represent positions. In the Cartesian system, points are fixed according to horizontal and vertical distances, while the polar system uses angles and distances from the origin.Here's how to graph the Cartesian point \((\sqrt{2}, -\sqrt{2})\):

• Locate the x-coordinate \(\sqrt{2}\) on the horizontal axis.
• Find the y-coordinate \(-\sqrt{2}\) on the vertical axis.
• Mark where these coordinates intersect to place the point.

In polar coordinates, the process involves angles and radial lines. You'd use the given radius and angle to determine position relative to the origin, moving counterclockwise from the positive x-axis.Understanding both systems' graphing procedures is crucial when plotting points from various coordinate types, helping visualize mathematical relationships and solutions seamlessly.