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Polar Coordinates

Polar Coordinates. Nate Long. Differences: Polar vs. Rectangular. POLAR. RECTANGULAR. (0,0) is called the pole Coordinates are in form (r, θ ). (0,0) is called the origin Coordinates are in form (x,y). How to Graph Polar Coordinates. Given: (3, л /3). Answer- STEP ONE.

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Polar Coordinates

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  1. Polar Coordinates Nate Long

  2. Differences: Polar vs. Rectangular POLAR RECTANGULAR • (0,0) is called the pole • Coordinates are in form (r, θ) • (0,0) is called the origin • Coordinates are in form (x,y)

  3. How to Graph Polar Coordinates • Given: (3, л/3)

  4. Answer- STEP ONE • Look at r and move that number of circles out • Move 3 units out (highlighted in red) 1 2 3

  5. Answer- STEP TWO • Look at θ- this tells you the direction/angle of the line • Place a point where the r is on that angle. • In this case, the angle is л/3 1 2 3

  6. Answer: STEP THREE • Draw a line from the origin through the point

  7. Converting Coordinates • Remember: The hypotenuse has a length of r. The sides are x and y. • By using these properties, we get that: x = rcosθ y=rsinθ tanθ=y/x r2=x2+y2 3, л/3 r y x

  8. CONVERT: Polar to Rectangle: (3, л/3) • x=3cos(л/3) x=3cos(60) 1.5 • y=3sin(л/3) x=3sin(60) 2.6 • New coordinates are (1.5, 2.6) ***x = rcosθ ***y=rsinθ tanθ=y/x r2=x2+y2

  9. CONVERT: Rectangular to Polar: (1, 1) (You could also find r by recognizing this is a 45-45-90 right triangle) • Find Angle: tanθ= y/x tanθ= 1 tan-1(1)= л/4 • Find r by using the equation r2=x2+y2 • r2=12+12 • r= √2 • New Coordinates are (√2, л/4)

  10. STEP ONE: Substitute into equation ***x = rcosθ ***r2=x2+y2 r2+4rcosθ=0 r + 4cosθ=0 (factor out r) Final Equation: r= -4cosθ ***x = rcosθ y=rsinθ tanθ=y/x ***r2=x2+y2

  11. Convert Equation to Polar: 2x+y=0 STEP TWO: Factor out r r(2cosθ + sinθ) = 0 graph of 2x+y=0 ***x = rcosθ ***y=rsinθ tanθ=y/x r2=x2+y2

  12. SYMMETRY: THINGS TO REMEMBER • When graphing, use these methods to test the symmetry of the equation

  13. Graphing Equations with Symmetry • GRAPH: r=2+3cosθ • ANSWER: STEP ONE: Make a Table and Choose Angles. Solve the equation for r.

  14. Graphing Equations with Symmetry • GRAPH: r=2+3cosθ • ANSWER: STEP TWO: Determine Symmetry Since the answer is the same, we know that this graph is symmetric along the polar axis

  15. Graph Answer: r=2+3cosθ л/3 л/6 2л 11л/6 5л/3 We know it is symmetrical through the polar axis 3л/2

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