convert the point (x,y,z)=(4, - 4, - 1) to cylindrical coordinates. give answers either as expressions, or…

convert the point (x,y,z)=(4, - 4, - 1) to cylindrical coordinates. give answers either as expressions, or decimals to two decimal places, with positive values for θ and r. (r,θ,z)= question help: video message instructor submit question

convert the point (x,y,z)=(4, - 4, - 1) to cylindrical coordinates. give answers either as expressions, or decimals to two decimal places, with positive values for θ and r. (r,θ,z)= question help: video message instructor submit question

Answer

Explanation:

Step1: Calculate the value of $r$

The formula for $r$ in cylindrical - coordinates is $r = \sqrt{x^{2}+y^{2}}$. Given $x = 4$ and $y=-4$, we have $r=\sqrt{4^{2}+(-4)^{2}}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\approx5.66$.

Step2: Calculate the value of $\theta$

The formula for $\theta$ is $\theta=\arctan(\frac{y}{x})$. Given $x = 4$ and $y=-4$, $\theta=\arctan(\frac{-4}{4})=\arctan(-1)$. Since the point $(x,y)=(4, - 4)$ is in the fourth - quadrant, and we want a positive value for $\theta$, $\theta = 2\pi-\frac{\pi}{4}=\frac{7\pi}{4}\approx5.497787\approx5.50$ (in radians).

Step3: The $z$ - value remains the same

The $z$ - value in cylindrical coordinates is the same as in Cartesian coordinates. So $z=-1$.

Answer:

$(4\sqrt{2},\frac{7\pi}{4},-1)\approx(5.66,5.50,-1)$