8 5 Polar Coordinates The rectangular coordinate system

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

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

Rectangular vs Polar Equations Rectangular equations are written in x and y Polar equations are written with variables r and x = r cos y = r sin x 2 + y 2 = r 2 tan = y Rectangular equations can be written in an equivalent polar form x Example 1: 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 ) Example 2: Convert x 2 + y 2 = 4 (equation of circle) to polar form r 2 = 4 r = 2 or r = -2

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

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) 90 degrees r 0 15 30 45 60 75 90 3 2. 6 1. 5 0 -1. 5 -2. 6 -3 180 degrees 270 degrees P. 387 in your text shows various types of polar graphs and associated equation forms.

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.

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

8. 6 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 = t 2 and y = 2 t + 3 for t in the interval [-3, 3] Graph these equations by making a Convert to rectangular form by t/x/y chart, then graphing points (x, y) Eliminating the parameter ‘t’ T x y Step 1: Solve 1 equation for t -3 9 -3 Step 2: Substitute ‘t’ into the -2 4 -1 ‘other’ equation -1 1 1 0 0 3 Y = 2 t + 3 t = (y – 3)/2 1 1 5 2 4 7 2 X = ((y – 3)/2) 3 9 9 X = (y – 3)2 4

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

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

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. Example 2: Write 8 – 8 i in trigonometric form. The reference angle Is 45 degrees so θ = 315 degrees.

Product Rule Example from your book 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

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

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