Five-Minute Check (over Lesson 9-2) Then/Now Key Concept: Convert Polar to Rectangular Coordinates Example 1: Polar Coordinates to Rectangular Coordinates Key Concept: Convert Rectangular to Polar Coordinates Example 2: Rectangular Coordinates to Polar Coordinates Example 3: Real-World Example: Conversion of Coordinates Example 4: Rectangular Equations to Polar Equations Example 5: Polar Equations to Rectangular Equations Lesson Menu

Convert between polar and rectangular coordinates. You used a polar coordinate system to graph points and equations. (Lessons 9-1 and 9-2) Convert between polar and rectangular coordinates. Convert between polar and rectangular equations. Then/Now

Key Concept 1

A. Find the rectangular coordinates for . Polar Coordinates to Rectangular Coordinates A. Find the rectangular coordinates for . x = r cos  Conversion formula y = r sin  = 2 cos r = 2 and  = = 2 sin = Simplify. = = 1 = Example 1

The rectangular coordinates of D are or approximately (1, 1.73). Polar Coordinates to Rectangular Coordinates The rectangular coordinates of D are or approximately (1, 1.73). Answer: Example 1

B. Find the rectangular coordinates for F(–5, 45°). Polar Coordinates to Rectangular Coordinates B. Find the rectangular coordinates for F(–5, 45°). For F(–5, 45°), r = –5 and  = 45°. x = r cos  Conversion formula y = r sin  = –5 cos 45° r = –5 and  = 45° = –5 sin 45° = Simplify. = = = Example 1

The rectangular coordinates of F are or approximately (–3.54, –3.54). Polar Coordinates to Rectangular Coordinates The rectangular coordinates of F are or approximately (–3.54, –3.54). Answer: Example 1

C. Find the rectangular coordinates for H(4, –240°). Polar Coordinates to Rectangular Coordinates C. Find the rectangular coordinates for H(4, –240°). For H(4, –240°), r = 4 and  = –240°. x = r cos  Conversion formula y = r sin  = 4 cos (–240°) r = 4 and  = –240° = 4 sin (–240°) = Simplify. = = –2 = Example 1

The rectangular coordinates of H are or approximately (–2, 3.46). Polar Coordinates to Rectangular Coordinates The rectangular coordinates of H are or approximately (–2, 3.46). Answer: Example 1

Find the rectangular coordinates for R(–8, 300°). B. C. D. Example 1

Key Concept 2

A. Find two pairs of polar coordinates for the point E(2, –4). Rectangular Coordinates to Polar Coordinates A. Find two pairs of polar coordinates for the point E(2, –4). For E(x, y) = (2, –4), x = 2 and y = –4. Because x > 0, use tan–1 to find . Conversion formula x = 2 and y = –4 Simplify. ≈ –1.11 or 5.18 Example 2

Answer: Sample Answer: E(4.47, –1.11) and E(4.47, 5.18) Rectangular Coordinates to Polar Coordinates One set of polar coordinates for E is (4.47, –1.11). Another representation that uses a positive -value is (4.47, –1.107 + 2π) or (4.47, 5.18), as shown. 5.18 Answer: Sample Answer: E(4.47, –1.11) and E(4.47, 5.18) Example 2

B. Find two pairs of polar coordinates for the point G(–2, –4). Rectangular Coordinates to Polar Coordinates B. Find two pairs of polar coordinates for the point G(–2, –4). For G(x, y) = (–2, –4), x = –2 and y = –4. Because x < 0, use to find . Conversion formula x = –2 and y = –4 Example 2

Rectangular Coordinates to Polar Coordinates Simplify. One set of polar coordinates for G is approximately (4.47, 4.25). Another representation that uses a negative r-value is (–4.47, 4.25 + π) or (–4.47, 7.39), as shown. Answer: G(4.47, 4.25) and G(–4.47, 7.39) Example 2

Find two pairs of polar coordinates for F(–5, –6) with the given rectangular coordinates. A. F(7.81, 0.69) or F(–7.81, 3.83) B. F(–7.81, 0.69) or F(7.81, 3.83) C. F(7.81, 0.88) or F(–7.81, 4.02) D. F(–7.81, 0.88) or F(7.81, 4.02) Example 2

x = r cos  Conversion formula y = r sin  Conversion of Coordinates A. ROBOTICS Refer to the beginning of the lesson. Suppose the robot is facing due east and its sensor detects an object at (3, 280°). What are the rectangular coordinates that the robot will need to calculate? x = r cos  Conversion formula y = r sin  = 3 cos 280o r = 3 and  = 280o = 3 sin 280o ≈ 0.52 Simplify. ≈ –2.95 Example 3

The object is located at the rectangular coordinates (0.52, –2.95). Conversion of Coordinates The object is located at the rectangular coordinates (0.52, –2.95). Answer: (0.52, –2.95) Example 3

Conversion of Coordinates B. ROBOTICS Refer to the beginning of the lesson. If a previously detected object has rectangular coordinates of (4, 9), what are the distance and angle measure of the object relative to the front of the robot? Conversion formula x = 4 and y = 9 Example 3

The object is located at the polar coordinates (9.85, 66.0°). Conversion of Coordinates ≈ 9.85 Simplify. ≈ 66.0° The object is located at the polar coordinates (9.85, 66.0°). Answer: (9.85, 66.0°) Example 3

HIDDEN TREASURE A crew is using radar to search for pirate treasure hidden under water. Suppose the boat is facing due east, and the radar gives the polar coordinates of the treasure as (8, 205o). What are the rectangular coordinates for the hidden treasure? A. (–2.56, –1.20) B. (–7.25, –3.38) C. (–7.25, 1.69) D. (2.56, 1.2) Example 3

(x + 2)2 + y 2 = 4 Original equation Rectangular Equations to Polar Equations A. Identify the graph of (x + 2) 2 + y 2 = 4. Then write the equation in polar form. Support your answer by graphing the polar form of the equation. The graph of (x + 2)2 + y 2 = 4 is a circle with radius 2 centered at (–2, 0). To find the polar form of this equation, replace x with r cos  and y with r sin . Then simplify. (x + 2)2 + y 2 = 4 Original equation (r cos  + 2)2 + (r sin )2 = 4 x = r cos  and y = r sin  Example 4

r 2 cos2  + 4r cos  + 4 + r 2 sin2  = 4 Multiply. Rectangular Equations to Polar Equations r 2 cos2  + 4r cos  + 4 + r 2 sin2  = 4 Multiply. r 2 cos2  + 4r cos  + r 2 sin2  = 0 Subtract 4 from each side. r 2 cos2  + r 2 sin2  = –4r cos  Isolate the squared terms. r 2(cos2  + sin2  ) = –4r cos  Factor. r 2(1) = –4r cos  Pythagorean Identity r = –4 cos  Divide each side by r. Example 4

Answer: circle; r = –4 cos  Rectangular Equations to Polar Equations The graph of this polar equation is a circle with radius 2 centered at (–2, 0). Answer: circle; r = –4 cos  Example 4

2(r cos )(r sin ) = 4 x = r cos  and y = r sin  Rectangular Equations to Polar Equations B. Identify the graph of 2xy = 4. Then write the equation in polar form. Support your answer by graphing the polar form of the equation. The graph of 2xy = 4 is a hyperbola with the x- and y-axes as asymptotes. 2xy = 4 Original equation 2(r cos )(r sin ) = 4 x = r cos  and y = r sin  2r 2 (cos )(sin ) = 4 Multiply. Example 4

Divide each side by 2(cos )(sin ) Rectangular Equations to Polar Equations Divide each side by 2(cos )(sin ) Double Angle Identity Reciprocal Identity The graph of the polar equation r 2 = 4 csc 2θ is a hyperbola with the x- and y-axes as the asymptotes. Example 4

Answer: hyperbola; r2 = csc 2 Rectangular Equations to Polar Equations Answer: hyperbola; r2 = csc 2 Example 4

Identify the graph of x = y 2. Then write the equation in polar form. A. circle; r = sin2  B. ellipse; r 2 = cos  C. hyperbola; r = tan  sin  D. parabola; r = cot  csc  Example 4

Find the tangent of each side. Polar Equations to Rectangular Equations A. Write the equation in rectangular form and then identify its graph. Support your answer by graphing the polar form of the equation. Original equation Find the tangent of each side. tan  = Multiply each side by x. Example 5

Polar Equations to Rectangular Equations The graph of the equation is a line through the origin with a slope of 1. Answer: line; y = x Example 5

Polar Equations to Rectangular Equations B. Write the equation r = 5 in rectangular form and then identify its graph. Support your answer by graphing the polar form of the equation. r = 5 Original equation r 2 = 25 Square each side. x 2 + y 2 = 25 r 2 = x 2 + y 2 The graph of the rectangular equation is a circle with radius 5 centered at (0, 0). Example 5

Polar Equations to Rectangular Equations Answer: circle; x 2 + y 2 = 25 Example 5

r = 2 sin  Original equation r 2 = 2r sin  Multiply each side by r. Polar Equations to Rectangular Equations C. Write the equation r = 2 sin θ in rectangular form and then identify its graph. Support your answer by graphing the polar form of the equation. r = 2 sin  Original equation r 2 = 2r sin  Multiply each side by r. x2 + y 2 = 2y r 2 = x 2 + y 2 and y = r sin  x2 + y2 – 2y = 0 Subtract 2y from each side. Because in standard form, x2 + (y – 1)2 = 1, you can identify the graph of this equation as a circle with radius 1 centered at (0, 1). Example 5

Answer: circle; x 2 + (y – 1)2 = 1 Polar Equations to Rectangular Equations Answer: circle; x 2 + (y – 1)2 = 1 Example 5

Write the equation r = 6 sin  in rectangular form and then identify its graph. A. x 2 + (y – 3)2 = 9; circle B. ; ellipse C. ; hyperbola D. x2 = 6y; parabola Example 5