Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in polar coordinates and find the corresponding rectangular coordinates for the point. $$\left(-3,-\frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

The rectangular coordinates for the point \((-3, -\pi/6)\) in polar coordinates are \((-sqrt{3}, 3/2) \).

Step 1: Analyzing Polar Coordinates

Polar coordinates are given as \((-3, -\pi/6)\). The negative radius means that the point is in the opposite direction of the angle. Since the angle is negative, we will go clockwise from the positive x-axis so the true angle for the point will be \((3, 5\pi/6)\) in standard position.

Step 2: Plotting in Polar Coordinates

Let's plot the point using the revised coordinates. From the origin (0,0), move along the positive x-axis till you reach the radial distance which is 3. Then rotate counterclockwise from there by an angle of \(5\pi/6\) radians. This will give the location of the point in polar coordinates.

Step 3: Converting Polar to Rectangular Coordinates

The formulas to convert polar coordinates to rectangular coordinates are \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Plug \(r = 3\) and \(\theta = 5\pi/6\) into these formulas to find the rectangular coordinates. Doing so gives \(x = 3\cos(5\pi/6) = -\sqrt{3}\) and \(y = 3\sin(5\pi/6) = 3/2\). So the rectangular coordinates of the point are \((-sqrt{3}, 3/2)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to determine the position of a point in the Cartesian plane. By using two values, typically labeled as \((x, y)\), they provide a simple grid-based method to locate points. The \(x\) value tells you how far to move horizontally:

• Positive \(x\) means move right.
• Negative \(x\) means move left.

The \(y\) value tells you how far to move vertically:

• Positive \(y\) means move up.
• Negative \(y\) means move down.

All points on this plane are defined in relation to the origin \((0, 0)\). Moving along these axes enables us to pinpoint any location precisely. Knowing how to read and use rectangular coordinates is foundational for mapping and graphical work.
Understanding this form of coordinates helps in visualizing geometrical shapes, solving equations, and making transformations into other coordinate forms.

Converting Between Polar and Rectangular

Converting between polar and rectangular coordinates involves swapping from one coordinate system to another. Polar coordinates \((r, \theta)\) are based on a radius \(r\) and an angle \(\theta\). For this conversion, you'll need the following formulas:

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

To use these formulas to find the rectangular coordinates:1. Identify the radius and angle in polar form.2. Plug them into the formulas to calculate \(x\) and \(y\).
For example, with \(r = 3\) and \(\theta = 5\pi/6\), the formulas yield \(x = 3\cos(5\pi/6) = -\sqrt{3}\) and \(y = 3\sin(5\pi/6) = 3/2\).
This process helps in changing polar plots into familiar Cartesian grids, making complex calculations more intuitive and visual interpretations easier.

Plotting Points in Polar Coordinates

Plotting points in polar coordinates can initially seem tricky but becomes straightforward with practice. These coordinates are defined by a distance from the origin and an angle from the polar axis (usually the positive \(x\)-axis).

• The first value, the radius \(r\), tells how far from the origin the point will be.
• The second value, the angle \(\theta\), specifies the direction from the positive x-axis, measured counterclockwise.

If \(r\) is negative, as in our example \((-3, -\pi/6)\), it means moving in the opposite direction of the angle. Here, converting to \((3, 5\pi/6)\) explains moving positively along the corrected angle.
Start plotting by:1. Moving along the initial direction of the positive \(x\)-axis up to radius length.2. Rotating counterclockwise by the angle given.
This approach offers a dynamic way to visualize points and angles beyond linear structures, thereby enriching spatial understanding.