1 / 14

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.

quiana
Download Presentation

8.5 Polar Coordinates

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related