What are the rectangular coordinates of the point whose cylindrical coordinates are $3,120°,5$?

• A. $1.5,-2.6,5$
• B. $1.5,2.6,5$
• C. $-2.6,1.5,5$
• D. $-1.5,2.6,5$

Answer: (D)

Verified step by step guidance

1. 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.
2. To convert cylindrical coordinates \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\), use the formulas: \[x = r \cdot \cos(\theta)\] \[y = r \cdot \sin(\theta)\] \[z = z\]
3. 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.
4. 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$.
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.