Problem 6 - Essential Calculus

Question

Use the angle feature of a graphing utility to find the rectangular coordinates for the point given in polar coordinates. Plot the point. $$ (-2,11 \pi / 6) $$

Step-by-Step Solution

Answer

The rectangular coordinates for the given point in polar coordinates (-2,11pi/6) are \(-sqrt(3), 1\).

Step 1: Understand the Problem

The point is given in polar coordinates (-2,11pi/6). Polar coordinates are represented as (r, theta) where r is the radius and theta is the angle. We have to convert these polar coordinates into rectangular coordinates (x, y). For this purpose we will use the formulas \(x = r * cos\(\theta)\) and \(y = r * sin\(\theta)\). Notice that \(r\) is negative.

Step 2: Convert to Rectangular Coordinates

First, we replace \(r\) and \(\theta\) with their given values in the formulas for \(x\) and \(y\). This gives \(x = -2 * cos(11pi/6)\) and \(y = -2 * sin(11pi/6)\). Simplifying these expressions gives us the rectangular coordinates.

Step 3: Calculate the Rectangular Coordinates

To find the exact values, recall that cos(11pi/6) is \(sqrt(3)/2\) and sin(11pi/6) is \(-1/2\). Substituting these values into the expressions gives \(x = -2* sqrt(3)/2 = -sqrt(3)\) and \(y = -2*(-1/2) = 1\). So the point in rectangular coordinates is \(-sqrt(3), 1\).