International Studies Charter School. Pre-Calculus Section 6-3

International Studies Charter School. Pre-Calculus Section 6-3
Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Objectives: Plot points in the polar coordinate system. Find multiple sets of polar coordinates for a given point. Convert a point from polar to rectangular coordinates. Convert a point from rectangular to polar coordinates. Convert an equation from rectangular to polar coordinates. Convert an equation from polar to rectangular coordinates.

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Plotting Points in the Polar Coordinate System The foundation of the polar coordinate system is a horizontal ray that extends to the right. The ray is called the polar axis. The endpoint of the ray is called the pole. A point P in the polar coordinate system is represented by an ordered pair of numbers 𝑟,𝜃 . We refer to the ordered pair 𝑃= 𝑟,𝜃 as the polar coordinates of P.

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Plotting Points in the Polar Coordinate System We refer to the ordered pair 𝑃= 𝑟,𝜃 as the polar coordinates of P. 𝑟 is a directed distance from the pole to P. 𝜃 is an angle from the polar axis to the line segment from the pole to 𝑃. This angle can be measured in degrees or radians. Positive angles are measured counterclockwise from the polar axis. Negative angles are measured clockwise from the polar axis.

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Plot the point with the following polar coordinates: 3, 315° Because 315° is a positive angle, draw 𝜃=315° counterclockwise from the polar axis. Because 𝑟 = 3 is positive, plot the point going out three units on the terminal side of 𝜃. 𝟑𝟏𝟓°

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Plot the point with the following polar coordinates: −2, 𝜋 Because π is a positive angle, Draw 𝜃=𝜋 counterclockwise from the polar axis. 𝝅 Because 𝑟 = –2 is negative, plot the point going out two units along the ray opposite the terminal side of 𝜃.

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Plot the point with the following polar coordinates: −1, − 𝜋 2 Because − 𝜋 2 is a negative angle, draw 𝜃=− 𝜋 2 clockwise from the polar axis. Because 𝒓 = –𝟏 is negative, plot the point going out one unit along the ray opposite the terminal side of − 𝝅 𝟐

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Multiple Representations of Points in the Polar Coordinate System

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find another representation of 5, 𝜋 4 in which r is positive and 𝟐𝝅<𝜽<𝟒𝝅. We want r positive and 𝟐𝝅<𝜽<𝟒𝝅. Add 2π to the angle and do not change r. 5, 𝜋 4 = 5, 𝜋 4 +𝟐𝝅 = 5, 𝜋 4 + 8𝜋 4 = 𝟓, 𝟗𝝅 𝟒

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find another representation of 5, 𝜋 4 in which r is positive and 𝝅<𝜽<𝟐𝝅. We want r negative and 𝝅<𝜽<𝟐𝝅. Add π to the angle and replace r with -r. 5, 𝜋 4 = −5, 𝜋 4 +𝝅 = −5, 𝜋 4 + 4𝜋 4 = −𝟓, 𝟓𝝅 𝟒

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find another representation of 5, 𝜋 4 in which r is positive and −𝟐𝝅<𝜽<𝟒𝝅. We want r positive and −𝟐𝝅<𝜽<𝟒𝝅. Subtract 2π to the angle and do not change r. 5, 𝜋 4 = 5, 𝜋 4 −𝟐𝝅 = 5, 𝜋 4 − 8𝜋 4 = 𝟓,− 𝟕𝝅 𝟒

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Relations between Polar and Rectangular Coordinates We now consider both polar and rectangular coordinates simultaneously. The polar axis coincides with the positive and the pole coincides with the origin. A point 𝑃, other than the origin, has rectangular coordinates 𝑥,𝑦 and polar coordinates 𝑟,𝜃 as indicated in the figure. We wish to find equations relating the two sets of coordinates. From the figure, we see that These relationships hold when 𝑃 is in any quadrant and when 𝑟>0 or 𝑟< 0.

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Relations between Polar and Rectangular Coordinates

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates To convert a point from polar coordinates 𝑟,𝜃 to rectangular coordinates (𝑥, 𝑦), use the formulas 𝒙=𝒓 𝐜𝐨𝐬 𝜽 or 𝒚=𝒓 𝐬𝐢𝐧 𝜽

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find the rectangular coordinates of the point with the following polar coordinates: (𝟑, 𝝅) 𝑟,𝜃 = 3,𝜋 𝑥=𝑟 cos 𝜃 =3 cos 𝜋 =3(−1) =−𝟑 𝑦=𝑟 sin 𝜃 =3 sin 𝜋 =3(0) =𝟎 The rectangular coordinates of (𝟑, 𝝅) are (–𝟑, 𝟎).

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find the rectangular coordinates of the point with the following polar coordinates: (−𝟏𝟎, 𝝅 𝟔 ) 𝑟,𝜃 = −10, 𝜋 6 =− =−10 cos 𝜋 6 =−𝟓 𝟑 𝑥=𝑟 cos 𝜃 =−6 1 2 =−10 sin 𝜋 6 𝑦=𝑟 sin 𝜃 =−𝟓 The rectangular coordinates of (−𝟏𝟎, 𝝅 𝟔 ) are (–𝟓 𝟑 , −𝟓).

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Point Conversion from Rectangular to Polar Coordinates Conversion from rectangular coordinates (𝑥, 𝑦) to polar coordinates (𝑟,𝜃), is a bit more complicated. Keep in mind that there are infinitely many representations for a point in polar coordinates. If the point (𝑥, 𝑦) lies in one of the four quadrants, we will use a representation in which is positive, and 𝜃 is the smallest positive angle with the terminal side passing through 𝑥,𝑦

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Point Conversion from Rectangular to Polar Coordinates These conventions provide the following procedure:

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find polar coordinates of the point whose rectangular coordinates are 1,− 3 Step 1 Plot the point (x, y). Step 2 Find r by computing the distance from the origin to (x, y). 𝑟= 𝑥 2 + 𝑦 2 = (1) 2 + − = 1+3 = 4 =𝟐

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find polar coordinates of the point whose rectangular coordinates are 1,− 3 Step 3 Find 𝜃 using 𝐭𝐚𝐧 𝜽= 𝒚 𝒙 with the terminal side of 𝜃 passing through (𝑥, 𝑦). = − 3 1 tan 𝜃 = 𝑦 𝑥 =− 𝟑 𝜽= 𝟓𝝅 𝟑

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find polar coordinates of the point whose rectangular coordinates are 1,− 3 One representation of 1,− 3 in polar coordinates is 𝒓,𝜽 = 𝟐, 𝟓𝝅 𝟑

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find polar coordinates of the point whose rectangular coordinates are −2,0 Step 1 Plot the point (x, y). Step 2 Find r by computing the distance from the origin to (x, y). 𝑟= 𝑥 2 + 𝑦 2 = (−2) = 4 =𝟐

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Find polar coordinates of the point whose rectangular coordinates are 1,2 Step 3 Find 𝜃 with 𝜃 lying on the same positive or negative axis as (𝑥, 𝑦). The point 2,0 is on the negative x-axis. Thus 𝜃 lies on the negative x-axis and 𝜃=𝜋. One representation of (−2,0) in polar coordinates is (𝟐,𝝅).

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Equation Conversion from Rectangular to Polar Coordinates

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Convert the following rectangular equation to a polar equation that expresses r in terms of 𝜃. 𝟑𝒙−𝒚=𝟔 The polar equation for 𝟑𝒙−𝒚=𝟔 is 𝟑𝒓 𝐜𝐨𝐬 𝜽−𝒓 𝐬𝐢𝐧 𝜽=𝟔 𝒓 𝟑 𝐜𝐨𝐬 𝜽− 𝐬𝐢𝐧 𝜽 =𝟔 𝒓= 𝟔 𝟑 𝐜𝐨𝐬 𝜽 − 𝐬𝐢𝐧 𝜽 𝒓= 𝟔 𝟑 𝐜𝐨𝐬 𝜽 − 𝐬𝐢𝐧 𝜽

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Convert the following rectangular equation to a polar equation that expresses r in terms of 𝜃. 𝒙 𝟐 + 𝒚+𝟏 𝟐 =𝟏 𝒓 𝒓+𝟐 sin 𝜽 =𝟎 𝒓 𝐜𝐨𝐬 𝜽 𝟐 + 𝒓 𝐬𝐢𝐧 𝜽 +𝟏 𝟐 =𝟏 𝒓=𝟎 𝒓+𝟐 sin 𝜽 =𝟎 𝒓 𝟐 cos 𝟐 𝜽 + 𝒓 𝟐 sin 𝟐 𝜽+𝟐𝒓 sin 𝜽+𝟏 =𝟏 𝒓=−𝟐 sin 𝜽 𝒓 𝟐 𝒄𝒐𝒔 𝟐 𝜽+ 𝒔𝒊𝒏 𝟐 𝜽 +𝟐𝒓 𝐬𝐢𝐧 𝜽 +𝟏=𝟏 𝒓 𝟐 𝟏 +𝟐𝒓 sin 𝜽 +𝟏=𝟏 The polar equation for 𝒙 𝟐 + 𝒚+𝟏 𝟐 =𝟏 is 𝒓=−𝟐 𝒔𝒊𝒏 𝜽 𝒓 𝟐 +𝟐𝒓 sin 𝜽 +𝟏=𝟏 𝒓 𝟐 +𝟐𝒓 sin 𝜽 =𝟎

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates Equation Conversion from Polar to Rectangular Coordinates When we convert an equation from polar to rectangular coordinates, our goal is to obtain an equation in which the variables are x and y rather than r and 𝜃. We use one or more of the following equations: To use these equations, it is sometimes necessary to square both sides, to use an identity, to take the tangent of both sides, or to multiply both sides by r.

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates a) Convert the polar equation to a rectangular equation in x and y: 𝒓=𝟒 The rectangular equation for is

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates 𝜽= 𝟑𝝅 𝟒 b) Convert the polar equation to a rectangular equation in x and y: The rectangular equation for is

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates c) Convert the polar equation to a rectangular equation in x and y: 𝒓=−𝟐 sec 𝜽 The rectangular equation for is

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates d) Convert the polar equation to a rectangular equation in x and y: 𝒓=𝟏𝟎 sin 𝜽 The rectangular equation for is

Mrs. Rivas International Studies Charter School. Pre-Calculus Section 6-3 Polar Coordinates d) Convert the polar equation to a rectangular equation in x and y: 𝒓=𝟏𝟎 sin 𝜽 The rectangular equation for 𝒓=𝟏𝟎 sin 𝜽 is 𝒙 𝟐 + 𝒚−𝟓 𝟐 𝒚−𝟓 𝟐 =𝟐𝟓

Mrs. Rivas Pg. 673 # 4, 6, 10, 12, 16, 20, 24, 26, 28, 30, 32, 34, 38, 40, 42, 46, 50, 54, 56, 58, 62, 64, 66, 70, 72, 74.

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International Studies Charter School. Pre-Calculus Section 6-3

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