Math Problem Statement

Express the point (4, 4√3, 0) in polar coordinates.

Solution

To express the point (4,43,0)(4, 4\sqrt{3}, 0) in polar coordinates, let's go through the following steps.

Step 1: Identify the coordinates

The point is given in Cartesian coordinates (x,y,z)(x, y, z) as: x=4,y=43,z=0x = 4, \quad y = 4\sqrt{3}, \quad z = 0

Step 2: Calculate rr

In polar coordinates, rr (the radial distance) is calculated as: r=x2+y2+z2r = \sqrt{x^2 + y^2 + z^2} Substitute x=4x = 4, y=43y = 4\sqrt{3}, and z=0z = 0:

= \sqrt{16 + 48} = \sqrt{64} = 8$$ ### Step 3: Calculate $$\theta$$ The angle $$\theta$$ in polar coordinates is given by: $$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$ Substitute $$x = 4$$ and $$y = 4\sqrt{3}$$: $$\theta = \tan^{-1}\left(\frac{4\sqrt{3}}{4}\right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \text{ radians}$$ ### Step 4: Determine $$z$$-axis component (if cylindrical) In this problem, since $$z = 0$$, the point lies in the $$xy$$-plane. In cylindrical coordinates, the $$z$$-coordinate remains as it is. So $$z = 0$$ remains the same in polar form. ### Final Answer The polar coordinates (or cylindrical coordinates) for the point $$(4, 4\sqrt{3}, 0)$$ are: $$(r, \theta, z) = (8, \frac{\pi}{3}, 0)$$ Would you like further details, or do you have any questions about this conversion? ### Related Questions 1. How would you convert a point with a non-zero $$z$$-coordinate to polar or cylindrical coordinates? 2. What are the steps to convert polar coordinates back to Cartesian coordinates? 3. How would the conversion change if the point were located in a different quadrant? 4. Can you express the same point in spherical coordinates? 5. How do you interpret negative radial distances in polar coordinates? ### Tip Always double-check your angle calculation to ensure it's in the correct quadrant, as $$\tan^{-1}$$ provides the principal value.

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Math Problem Analysis

Mathematical Concepts

Coordinate Transformation
Polar Coordinates
Trigonometry

Formulas

r = √(x^2 + y^2 + z^2)
θ = tan⁻¹(y/x)

Theorems

Pythagorean Theorem

Suitable Grade Level

Grades 11-12

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