A polar point satisfies . A negative reverses the named ray through the pole, while distance from the pole is . The compass artwork is schematic rather than a numbered polar grid.
What a strong response shows
The angle chooses a ray; a negative radius reverses the plotting direction along the same line.
Use and with the SIGNED radius.
Equivalent polar coordinates can change the angle by full turns or reverse radius with a half-turn.
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What You Need to Know
A polar point is located by a directed angle and a signed distance from the pole. The same point can have many polar descriptions.
Sketch the ray first, then move the signed radius along it or the opposite ray.
Locate the angle from the positive horizontal axis, then place the point at the stated radial distance.
Treating a negative radius as an invalid point.
What does a negative radius do?
Learning Objectives
3.13.A — Determine the location of a point in the plane using both rectangular and polar coordinates.
For a coordinate conversion, keep angle units consistent and use the correct coordinate formulas.
. Identify radius and angle. . Draw the polar ray. . Convert or plot. . Check the quadrant.
Using a degree angle inside a radian formula without conversion.
Which coordinate uses cosine?
Key Concepts
Polar coordinates use direction and distance; rectangular coordinates use horizontal and vertical displacement.
Polar tells how far and which direction; rectangular tells left-right and up-down.
Project the radial segment onto the axes to obtain horizontal and vertical components.
Adding the angle to the radius as if they were the same kind of quantity.
Which part of a polar pair is an angle?
Relevant Representations
A polar grid and a rectangular plane show the same point with different measurements.
| From | To | Relationship |
|---|---|---|
| polar | rectangular | horizontal projection and vertical projection |
| rectangular | polar | distance and direction |
| polar | polar equivalent | add a full turn or reverse radius with a half-turn |
Use a grid for geometry and formulas for conversion.
Draw the ray, mark the radius, then drop perpendicular guides to read rectangular coordinates.
Placing a negative radius on the named ray instead of the opposite ray.
Which guide helps you read the x-coordinate?
Essential Relationships or Formulas
Projection formulas connect polar and rectangular coordinates.
Use the quadrant to choose the correct direction for the angle.
Compute horizontal and vertical components, then check that their distance from the origin equals the absolute value of the signed radius.
Using the principal inverse tangent without quadrant correction.
How can the radius equation verify a conversion?
Worked Examples
A sketch prevents most polar-sign errors.
Plot before calculating the final angle.
For a negative radius, reverse the ray by one half-turn, plot the point, then convert the resulting rectangular coordinates.
Keeping the original ray after changing the radius sign.
Where does a negative radial distance point?
Common Mistakes
Polar errors come from sign, quadrant, angle-unit, and multiple-representation issues.
Check ray, sign, quadrant, and units in that order.
Compare the plotted point with the rectangular signs and the radius length.
Reporting a correct distance with the wrong direction.
What two features identify a polar point?
AP Exam Tips
Include a sketch or quadrant statement when converting coordinates.
Use the Questionbank set below to test this skill.
State the angle unit and use an exact form when available.
Show the conversion, quadrant check, and radius check.
Giving only calculator decimals for an exact-coordinate prompt.
Can your point be plotted from your answer alone?
Can you transfer the method?
Convert to rectangular coordinates, and name the actual quadrant of the point.
Compare your reasoning
, . The point is in Quadrant IV even though the parameter angle's ray lies in Quadrant II.
Work independently, then compare the mathematical evidence in your answer.