Question

Express the following polar coordinates in Cartesian coordinates. $$(4,5 \pi)$$

Short answer

Question: Convert the polar coordinates (4, 5π) to Cartesian coordinates. Answer: The Cartesian coordinates are (4, 0).

Step-by-step solution

Identify the values of r and θ

In the given polar coordinates \((4,5 \pi)\), we have: \(r = 4\) \(\theta = 5 \pi\)

Convert r and θ to x and y using conversion formulas

We will use the conversion formulas to find the Cartesian coordinates: \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)

Calculate the x-coordinate

Using the conversion formula for x-coordinate and the given values of r and θ, we get: \(x = 4 \cos(5\pi)\) Note that \(\cos(5\pi) = 1\) because cosine is positive for the even multiples (2, 4, 6...) of \(\pi\). So, \(x= 4 \cdot 1 =4\)

Calculate the y-coordinate

Using the conversion formula for y-coordinate and the given values of r and θ, we get: \(y = 4 \sin(5 \pi)\) Note that \(\sin(5 \pi) = 0\) because sine is zero for all integer multiples of \(\pi\) (0, 1, 2...). So, \(y= 4 \cdot 0 =0\)

Final Answer

Therefore, the Cartesian coordinates of the given point \((4,5 \pi)\) in polar coordinates is \((4,0)\).