Questions: Convert the polar coordinates S(16,135°) into rectangular coordinates. Do not use the polar-conversion function on your calculator. A. (-8 sqrt(2),-8 sqrt(2)) B. (-8 sqrt(2), 8 sqrt(2)) C. (-8 sqrt(3), 8) D. (8 sqrt(3), 8) E. (8 sqrt(2), 8 sqrt(2))

Solution Steps

To convert polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \), use the formulas: \[ x = r \cdot \cos(\theta) \] \[ y = r \cdot \sin(\theta) \]

Given \( S\left(16, 135^{\circ}\right) \), substitute \( r = 16 \) and \( \theta = 135^{\circ} \) into the formulas: \[ x = 16 \cdot \cos(135^{\circ}) \] \[ y = 16 \cdot \sin(135^{\circ}) \]

Using the unit circle: \[ \cos(135^{\circ}) = -\frac{\sqrt{2}}{2} \] \[ \sin(135^{\circ}) = \frac{\sqrt{2}}{2} \]

Substitute the values of \( \cos(135^{\circ}) \) and \( \sin(135^{\circ}) \) into the formulas: \[ x = 16 \cdot \left(-\frac{\sqrt{2}}{2}\right) = -8\sqrt{2} \] \[ y = 16 \cdot \left(\frac{\sqrt{2}}{2}\right) = 8\sqrt{2} \]

The rectangular coordinates are \( (-8\sqrt{2}, 8\sqrt{2}) \).

Final Answer