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8.5 Polar Coordinates

90 degrees. 180 degrees. 0 degrees. 270 degrees. 8.5 Polar Coordinates. The rectangular coordinate system (x/y axis) works in 2 dimensions with each point having exactly one representation. A polar coordinate system allows for the rotation and repetition of points.

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8.5 Polar Coordinates

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  1. 90 degrees 180 degrees 0 degrees 270 degrees 8.5 Polar Coordinates The rectangular coordinate system (x/y axis) works in 2 dimensions with each point having exactly one representation. A polar coordinate system allows for the rotation and repetition of points. Each point has infinitely many representations. A polar coordinate point is represented by an ordered pair (r, ) r  Examples to try (3, 30°) (2, 135) (-2, 30°) (-1, -45)

  2. y x 90 degrees 180 degrees 0 degrees Converting Coordinates: Polar to/from Rectangular P(x,y) r  Example: Polar to Rectangular Polar Point: (2, 30º) X = 2 cos 30 = 3 Y = 2 sin 30 = 1 Rectangular point: (3 , 1) Example: Rectangular to Polar Rextangular Point: (3, 5) 32 + 52 = r2  r = 34 tan  = 5/3   = 59º Polar point: (34 , 59º) x = r cos y = r sin  x2 + y2 = r2 tan  = y x Note: You can also convert rectangular Equations to polar equations and vice versa.

  3. x = r cos y = r sin  x2 + y2 = r2 tan  = y x Rectangular vs Polar Equations Rectangular equations are written in x and y Polar equations are written with variables r and  Rectangular equations can be written in an equivalent polar form Example1: Convert y = x - 3 (equation of a line) to polar form.  x – y = 3  (r cos) – (r sin ) = 3  r (cos - sin ) = 3  r = 3/(cos - sin ) Example2:Convert x2 + y2 = 4 (equation of circle) to polar form  r2 = 4  r = 2 or r = -2

  4. x = r cos y = r sin  x2 + y2 = r2 tan  = y x Rectangular vs Polar Equations Rectangular equations are written in x and y Polar equations are written with variables r and  Polar equations can be written in an equivalent rectangular form Example1: Convert to rectangular form.  r + rsinθ = 4  r + y = 4 = 3    x2 + y2 = (4 – y)2  x2 = -y2 + 16 -8y + y2 x2 = 16 -8y x2 – 16 = -8y y = - (1/8) x2 + 2

  5. 90 degrees 180 degrees 0 degrees 270 degrees Graphing Polar Equations To Graph a polar equation, Make a  / r chart for until a pattern apppears. Then join the points with a smooth curve. Example: r = 3 cos 2 (4 leaved rose)  r 0 15 30 45 60 75 90 3 2.6 1.5 0 -1.5 -2.6 -3 P. 387 in your text shows various types of polar graphs and associated equation forms.

  6. Graphing Polar Equations To Graph a polar equation, Make a  / r chart for until a pattern apppears. Then join the points with a smooth curve. Example: r = 3 cos 2 (4 leaved rose)  r 0 15 30 45 60 75 90 3 2.6 1.5 0 -1.5 -2.6 -3 P. 387 in your text shows various types of polar graphs and associated equation forms.

  7. Classifying Polar Equations • Circles and Lemniscates • Limaçons 2n leaves if n is even n≥ 2 and n leaves if n is odd • Rose Curves

  8. 8.6 Parametric Equations Parametric Equations are sometimes used to simulate ‘motion’ x = f(x) and y = g(t) are parametric equations with parameter, t when they Define a plane curve with a set of points (x, y) on an interval I. Example: Let x = t2 and y = 2t + 3 for t in the interval [-3, 3] Graph these equations by making a t/x/y chart, then graphing points (x,y) T x y -3 9 -3 -2 4 -1 -1 1 1 0 0 3 1 1 5 2 4 7 3 9 9 Convert to rectangular form by Eliminating the parameter ‘t’ Step 1: Solve 1 equation for t Step 2: Substitute ‘t’ into the ‘other’ equation Y = 2t + 3  t = (y – 3)/2 X = ((y – 3)/2)2 X = (y – 3)2 4

  9. Application: Toy Rocket • A toy rocket is launched from the ground with velocity 36 feet per second at an angle of 45° with the ground. Find the rectangular equation that models this path. What type of path does the rocket follow? The motion of a projectile (neglecting air resistance) can be modeled by for t in [0, k]. Since the rocket is launched from the ground, h = 0. The parametric equations determined by the toy rocket are Substitute from Equation 1 into equation 2: A Parabolic Path

  10. 8.2 & 8.3 Complex Numbers • Graphing Complex Numbers: • Use x-axis as ‘real’ part • Use y-axis as ‘imaginary’ part • Trig/Polar Form of Complex Numbers: • Rectangular form: a + bi • Polar form: r (cos  + isin ) are any two complex numbers, then Quotient Rule Product Rule

  11. Examples of Polar Form Complex Numbers • Trig/Polar Form of Complex Numbers: • Rectangular form: a + bi • Polar form: r (cos + isin ) Example 1: Express 10(cos 135° + i sin 135°) in rectangular form. Example2: Write 8 – 8iin trigonometric form. The reference angle Is 45 degrees so θ = 315 degrees.

  12. Product Rule Example from your book Find the product of 4(cos 120° + isin 120°) and 5(cos 30° + isin 30°). Write the result in rectangular form. Product Rule

  13. Quotient Rule Example from your Book • Find the quotient Note: CIS 45◦ is an abbreviation For (cos 45◦+ isin 45◦) Quotient Rule

  14. 8.4 De Moivre’s Theorem is a complex number, then Example: Find (1 + i3)8 and express the result in rectangular form 1st, express in Trig Form: 1 + i3 = 2(cos 60 + i sin 60) Now apply De Moivre’s Theorem: 480° and 120° are coterminal. Rectangular form

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