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- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_1350%3A_Precalculus_Part_I/10%3A_Further_Applications_of_Trigonometry/10.03%3A_Polar_CoordinatesWhen we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and ...When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled (r,θ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
- https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/06%3A_The_Polar_System/6.01%3A_Polar_CoordinatesThe polar grid is scaled as the unit circle with the positive \(x\)-axis now viewed as the polar axis and the origin as the pole. To plot a point in the form \((r,\theta)\), \(\theta>0\), move in a co...The polar grid is scaled as the unit circle with the positive \(x\)-axis now viewed as the polar axis and the origin as the pole. To plot a point in the form \((r,\theta)\), \(\theta>0\), move in a counterclockwise direction from the polar axis by an angle of \(\theta\), and then extend a directed line segment from the pole the length of \(r\) in the direction of \(\theta\).
- https://math.libretexts.org/Courses/Rio_Hondo/Math_175%3A_Plane_Trigonometry/06%3A_The_Polar_System/6.03%3A_Converting_Between_Systems\[\begin{aligned} r &=\sin(2\theta) && \text{Use the double angle identity for sine.} \\[4pt] r &=2 \sin \theta \cos \theta && \text{Use }\cos \theta=\dfrac{x}{r} \text{ and } \sin \theta =\dfrac{y}{r...\[\begin{aligned} r &=\sin(2\theta) && \text{Use the double angle identity for sine.} \\[4pt] r &=2 \sin \theta \cos \theta && \text{Use }\cos \theta=\dfrac{x}{r} \text{ and } \sin \theta =\dfrac{y}{r}. \\ r&=2 \left(\dfrac{x}{r}\right)\left(\dfrac{y}{r}\right) && \text{ Simplify.} \\[4pt] r &= \dfrac{2xy}{r^2} && \text{Multiply both sides by }r^2. \\[4pt] r^3 &=2xy \\[4pt] {(x^2+y^2)}^3 &=2xy && \text{As }x^2+y^2 =r^2, r=\sqrt{x^2+y^2}. \end{aligned}\]
- https://math.libretexts.org/Courses/Reedley_College/Trigonometry/04%3A_Further_Applications_of_Trigonometry/4.03%3A_Polar_CoordinatesWhen we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and ...When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled (r,θ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
- https://math.libretexts.org/Workbench/Algebra_and_Trigonometry_2e_(OpenStax)/10%3A_Further_Applications_of_Trigonometry/10.04%3A_Polar_CoordinatesWhen we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and ...When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled (r,θ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
- https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_(OpenStax)/10%3A_Further_Applications_of_Trigonometry/10.03%3A_Polar_CoordinatesWhen we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and ...When we think about plotting points in the plane, we usually think of rectangular coordinates (x,y) in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled (r,θ) and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
