2. 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,  ) 0 degrees 90 degrees 180 degrees 270 degrees r  Examples to try (3, 30°) (2, 135  ) (-2, 30°) (-1, -45  )

3. Converting Coordinates: Polar to/from Rectangular 0 degrees 90 degrees 180 degrees x y P(x,y) r  x = r cos  y = r sin  x 2 + y 2 = r 2 tan  = y x Note: You can also convert rectangular Equations to polar equations and vice versa. 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) 3 2 + 5 2 = r 2  r =  34 tan  = 5/3   = 59º Polar point: (  34, 59º)

4. 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 x = r cos  y = r sin  x 2 + y 2 = r 2 tan  = y x 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 x 2 + y 2 = 4 (equation of circle) to polar form  r 2 = 4  r = 2 or r = -2

5. 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 x = r cos  y = r sin  x 2 + y 2 = r 2 tan  = y x Example1: Convert to rectangular form.  r + rsinθ = 4  r + y = 4 = 3   x 2 + y 2 = (4 – y) 2  x 2 = -y 2 + 16 -8y + y 2 x 2 = 16 -8y x 2 – 16 = -8y y = - (1/8) x 2 + 2

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.  r Example: r = 3 cos 2  (4 leaved rose) 0 15 30 45 60 75 90 3 2.6 1.5 0 -1.5 -2.6 -3 0 degrees 90 degrees 180 degrees 270 degrees P. 387 in your text shows various types of polar graphs and associated equation forms.

7. 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.  r Example: r = 3 cos 2  (4 leaved rose) 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.

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

9. 8.6 Parametric Equations 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 = t 2 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 Parametric Equations are sometimes used to simulate ‘motion’

10. 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. Application: Toy Rocket The parametric equations determined by the toy rocket are Substitute from Equation 1 into equation 2: A Parabolic Path

11. 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 Product Rule Quotient Rule

12. 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 – 8i in trigonometric form. The reference angle Is 45 degrees so θ = 315 degrees.

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

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

15. 8.4 De Moivre’s Theorem is a complex number, then Example: Find (1 + i  3) 8 and express the result in rectangular form 1 st, 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