3.5 Describing how angles and radii change with respect to each other in a polar graph

• How are points located on a plane in the polar coordinate system?

  Distance and angle
• In polar coordinates, $r$ represents the distance from the origin
• The angle $\theta$ in polar coordinates is measured clockwise from the positive x-axis.


  False
• What is the formula to convert polar coordinates to x-coordinates?

  x = r \cos(\theta)</latex>
• To convert rectangular coordinates to polar coordinates, the formula for $r$ is $r =$ \sqrt{x^{2} + y^{2}}
• Match the coordinate system feature with its description:

  Origin ↔️ Reference point in rectangular coordinates

  Polar axis ↔️ Reference line in polar coordinates

  Euclidean geometry ↔️ Primary use of rectangular coordinates

  Spiral shapes ↔️ Primary use of polar coordinates
• In polar coordinates, as the angle increases, the radius always stays constant.

  False
• What shape is traced in a polar graph when the angle changes and the radius stays constant?

  Radial line
• Steps to plot a polar coordinate

  1️⃣ Start at the pole

  2️⃣ Move along the polar axis to the radius distance

  3️⃣ Measure the angle counterclockwise from the polar axis
• What determines the direction of a point from the pole in polar coordinates?

  Angle $\theta$
• Changing the angle in polar coordinates moves the point along a straight line.

  False
• In polar coordinates, if the angle stays constant and the radius increases, the point moves along a radial direction.
• What is the starting point for plotting polar coordinates?

  Pole
• When plotting polar coordinates, you first measure the angle and then the radius.

  False
• What does the radius $r$ represent in polar coordinates?


  Distance from the pole
• When the angle $\theta$ increases and the radius $r$ stays constant, the point pivots around the pole
• Changing the radius in polar coordinates moves the point along a straight line at a constant angle.
• Steps to plot a polar coordinate $(r, \theta)$:


  1️⃣ Start at the pole

  2️⃣ Move a distance $r$ along the polar axis

  3️⃣ Measure the angle $\theta$ counterclockwise
• What are the polar coordinates of the point $(3, \frac{\pi}{4})$?


  (3, \(\frac{\pi}{4}\))
• To convert (3, \frac{\pi}{4})</latex> to rectangular coordinates, $x$ is calculated as $r \cos(\theta)$, which equals \frac{3\sqrt{2}}{2}
• Match the polar graph characteristic with its description:

  Shape ↔️ Can be circles, spirals, roses

  Symmetry ↔️ Around x-axis, y-axis, or pole

  Intercepts ↔️ Graph intersections with axes
• What does a cardioid graph in polar coordinates look like?

  A heart
• In a polar graph, increasing the angle $\theta$ while the radius $r$ stays constant results in a circular arc.


  False
• When the angle $\theta$ decreases and the radius $r$ stays constant, the point pivots clockwise around the pole
• What shape is traced when $r =$$3$ and $\theta$ varies from $0$ to $2\pi$?


  A circle
• Match the feature with the correct coordinate system:

  Coordinates $(x, y)$ ↔️ Rectangular Coordinates

  Reference Line: Polar Axis ↔️ Polar Coordinates

  Reference Point: Origin ↔️ Both Coordinate Systems
• To convert the polar coordinate $(4, \frac{\pi}{3})$ to rectangular coordinates, $y$ is calculated as $r \sin(\theta)$, which equals 2\sqrt{3}
• What is the polar coordinate system used for?

  Locating points using distance and angle
• In polar coordinates, a point $P$ is described as $(r, \theta)$, where $r$ is the distance
• The angle θ in polar coordinates is measured clockwise from the positive x-axis.

  False
• What is the formula to convert polar coordinates to rectangular coordinates for $x$?


  $x =$$r \cos(\theta)$
• The formula to convert rectangular coordinates to polar coordinates for $\theta$ is $\theta =$$\arctan(\frac{y}{x})$, where $\arctan$ is the inverse tangent
• Match the coordinate type with its primary use:

  Rectangular Coordinates ↔️ Euclidean geometry

  Polar Coordinates ↔️ Circular and spiral shapes
• In polar coordinates, the angle θ and radius r have a direct relationship that defines the location of a point.
• Steps to plot a polar coordinate $(r, \theta)$


  1️⃣ Start at the pole (origin)

  2️⃣ Move a distance r from the pole along the polar axis

  3️⃣ Measure the angle θ counterclockwise from the polar axis
• What is the first step in plotting a polar coordinate?

  Start at the pole
• To plot a polar coordinate $(r, \theta)$, you start at the pole
• Match the characteristic with its description in polar graphs:

  Shape ↔️ Circles, spirals, lemniscates, roses

  Symmetry ↔️ Around the x-axis, y-axis, or pole

  Intercepts ↔️ Points where the graph crosses axes

  Maximum Values ↔️ Greatest distance from the pole
• What type of graph is represented by the equation $r =$$a \cos(\theta)$?


  Circle
• A cardioid graph is symmetric about the x-axis.