Chapter 7: Additional Applications of Integration

Section 7.1: Polar Coordinates

Example 7.1.4

$$ Convert the polar point $\left(-2\,3\right)$ to Cartesian coordinates. $$ Solution $$ According to the note at the end of the Introduction, a negative radius is made positive, but the angle is increased by $\mathrm{\π}$.   •  Write the expression for $x$. •  Context Panel: Evaluate and Display Inline •  Context Panel: Approximate≻5 (digits) $2 \cos \left(3\+\mathrm{\π}\right)$ = $-2\cos \left(3\right)$$\stackrel{\text{at 5 digits}}{\to }$$1.9800$$$ •  Write the expression for $y$. •  Context Panel: Evaluate and Display Inline •  Context Panel: Approximate≻5 (digits) $2 \sin \left(3\+\mathrm{\π}\right)$ = $-2\sin \left(3\right)$$\stackrel{\text{at 5 digits}}{\to }$$-0.28224$$$ $$ From Example 7.1.3, the polar point $\left(2\,3\right)$ is in the second quadrant. The polar point $\left(-2\,3\right)$ is a reflection of $\left(2\,3\right)$ through the origin, and is therefore in the fourth quadrant.

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