8.1: Plotting and Equivalent Representations | Precalculus

How it works

Polar coordinates give a signed distance and direction. Adding full turns preserves the point. Negating the radius reverses direction, so add pi to the angle when changing the sign of the radius.

(r,\theta)\equiv(r,\theta+2k\pi)\equiv(-r,\theta+(2k+1)\pi)

Worked examples

Example 1

Give two equivalent coordinates for $\displaystyle (2,\pi/3)$ .

Examples are $\displaystyle (2,7\pi/3)$ and $\displaystyle (-2,4\pi/3)$ .

Example 2

Plot $\displaystyle (-3,0)$ in Cartesian coordinates.

A negative radius moves opposite the positive x-direction, giving $\displaystyle (-3,0)$ in Cartesian coordinates.

Problems

Problem 8.1.1

Convert the polar point $\displaystyle (4,\pi/3)$ to Cartesian coordinates.

A radius of length 4 makes an angle pi/3 with the positive horizontal axis. — Use x = r cos(theta) and y = r sin(theta).

Show worked solution

$\displaystyle x=4\cos(\pi/3)=2$ and $\displaystyle y=4\sin(\pi/3)=2\sqrt{3}$ .

A radius of length 4 makes an angle pi/3 with the positive horizontal axis. The solution adds labeled relationships. — Cartesian coordinates are (2, 2 sqrt(3)).

Problem 8.1.2

Give an equivalent representation of $\displaystyle (3,\pi/4)$ with negative radius.

Show worked solution

One answer is $\displaystyle (-3,5\pi/4)$ .

Problem 8.1.3

Are $\displaystyle (2,\pi/2)$ and $\displaystyle (-2,3\pi/2)$ the same point?

Show worked solution

Yes. Both convert to $\displaystyle (0,2)$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 8.1.1

Problem 8.1.2

Problem 8.1.3

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