Questions: Convert these points from polar to Cartesian 5. (8, 3π/4)

Transcript text: Convert these points from polar to Cartesi 5. $\left(8, \frac{3 \pi}{4}\right)$

Solution

Solution Steps

Step 1: Given Polar Coordinates

The polar coordinates provided are $ (r, \theta) = \left(8, \frac{3\pi}{4}\right) $.

Step 2: Calculate $ x $

Using the formula for converting polar to Cartesian coordinates, we calculate $ x $: \[ x = r \cos(\theta) = 8 \cos\left(\frac{3\pi}{4}\right) \] Since $ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} $, we have: \[ x = 8 \left(-\frac{\sqrt{2}}{2}\right) = -4\sqrt{2} \]

Step 3: Calculate $ y $

Next, we calculate $ y $ using the formula: \[ y = r \sin(\theta) = 8 \sin\left(\frac{3\pi}{4}\right) \] Since $ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} $, we find: \[ y = 8 \left(\frac{\sqrt{2}}{2}\right) = 4\sqrt{2} \]

Step 4: Final Cartesian Coordinates

The Cartesian coordinates corresponding to the polar point $ \left(8, \frac{3\pi}{4}\right) $ are: \[ (x, y) = \left(-4\sqrt{2}, 4\sqrt{2}\right) \]