Problem 28 - 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. $$(4,11 \pi / 9)$$

Step-by-Step Solution

Answer

The rectangular coordinates of the point \((4, \frac{11\pi}{9})\) in polar coordinates are approximately \((x,y)\), where \(x\) and \(y\) are the calculated values from Step 2 and Step 3, respectively.

Step 1: Set Up Conversion Formulas

The first step is to apply the conversion formulas \(x = r\cos(\theta)\) for the x-coordinate and \(y = r\sin(\theta)\) for the y-coordinate. Here, \(r = 4\) and \(\theta = \frac{11\pi}{9}\).

Step 2: Determine X-Coordinate

The x-coordinate is found by substituting the given polar coordinates into the x-coordinate formula: \(x = 4\cos(\frac{11\pi}{9})\). Now, using a calculator, find the value of this calculation and round to two decimal places.

Step 3: Determine Y-Coordinate

The y-coordinate is found by substituting the given polar coordinates into the y-coordinate formula: \(y = 4\sin(\frac{11\pi}{9})\). Now, using a calculator, find the value of this calculation and round to two decimal places.