Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(2,2 \pi / 9)$$

Step-by-Step Solution

Answer

The rectangular coordinates of the polar point given as \((2, 2 \pi / 9)\) are approximately \((1.72, 0.76)\), given that the results are rounded to two decimal places.

Step 1: Set Up the Equations

Identify the given polar coordinates \(r\), the radial coordinate, and \(\theta\), the angular coordinate. In this case, \(r = 2\) and \(\theta = 2 \pi / 9\). Now, set up the equations using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) to convert these polar coordinates into rectangular coordinates.

Step 2: Calculate x-coordinate

To find the x-coordinate, substitute the given values into the equation \(x = r \cos(\theta)\). So, we get \(x = 2 \cos(2 \pi / 9)\). Now, use a calculator to find the numerical value of \(x\) and round your result to two decimal places.

Step 3: Calculate y-coordinate

To calculate the y-coordinate, substitute the given values into the equation \(y = r \sin(\theta)\). Therefore, \(y = 2 \sin(2 \pi / 9)\). To get a numerical value for \(y\), again use a calculator and round your result to two decimal places.