Textbook Question
In Exercises 81–86, solve each equation in the complex number system. Express solutions in polar and rectangular form.
x⁴ + 16i = 0
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9. Polar Equations
Polar Coordinate System
Multiple Choice
What are the rectangular coordinates of the point whose cylindrical coordinates are ?
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Verified step by step guidance1
Recall that cylindrical coordinates are given as \((r, \theta, z)\), where \(r\) is the radial distance from the \(z\)-axis, \(\theta\) is the angle measured in degrees (or radians) from the positive \(x\)-axis in the \(xy\)-plane, and \(z\) is the height above the \(xy\)-plane.
To convert cylindrical coordinates \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\), use the formulas:
\[x = r \cdot \cos(\theta)\]
\[y = r \cdot \sin(\theta)\]
\[z = z\]
Substitute the given values \(r = 3\), \(\theta = 120^\circ\), and \(z = 5\) into the formulas. Remember to convert the angle \(\theta\) to radians if your calculator requires it, or use the degree mode directly.
Calculate \(x\) by evaluating \(3 \cdot \cos(120^\circ)\) and calculate \(y\) by evaluating \(3 \cdot \sin(120^\circ)\). Keep the \(z\) coordinate the same as given, which is \(5\).
Combine the results to write the rectangular coordinates as \((x, y, z)\), which will give you the point in rectangular form corresponding to the original cylindrical coordinates.
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