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Find the exact values of the six trigonometric functions of θ. B. C. D. 5–Minute Check 1
Find the exact values of the six trigonometric functions of θ. B. C. D. 5–Minute Check 1
Graph points with polar coordinates. Graph simple polar equations. You drew positive and negative angles given in degrees and radians in standard position. (Lesson 4-2) Graph points with polar coordinates. Graph simple polar equations. Then/Now
polar coordinate system pole polar axis polar coordinates polar equation polar graph Vocabulary
Graph Polar Coordinates A. Graph S(1, 200°). Because = 200o, sketch the terminal side of a 200o angle with the polar axis as its initial side. Because r = 1, plot a point 1 unit from the pole along the terminal side of this angle. Answer: Example 1
Graph Polar Coordinates A. Graph S(1, 200°). Because = 200o, sketch the terminal side of a 200o angle with the polar axis as its initial side. Because r = 1, plot a point 1 unit from the pole along the terminal side of this angle. Answer: Example 1
Graph Polar Coordinates B. Graph . Because , sketch the terminal side of a angle with the polar axis as its initial side. Because r is negative, extend the terminal side of the angle in the opposite direction and plot a point 2 units from the pole along this extended ray. Example 1
Graph Polar Coordinates Answer: Example 1
Graph Polar Coordinates Answer: Example 1
Graph Polar Coordinates C. Graph M(4, –90°). Because = –90o, sketch the terminal side of a –90o angle with the polar axis as its initial side. Because r = 4, plot a point 4 units from the pole along the terminal side of this angle. Answer: Example 1
Graph Polar Coordinates C. Graph M(4, –90°). Because = –90o, sketch the terminal side of a –90o angle with the polar axis as its initial side. Because r = 4, plot a point 4 units from the pole along the terminal side of this angle. Answer: Example 1
Graph H(3, 120o). A. B. C. D. Example 1
Graph H(3, 120o). A. B. C. D. Example 1
Graph Points on a Polar Grid A. Graph on a polar grid. Because , sketch the terminal side of a angle with the polar axis as its initial side. Because r = 3, plot a point 3 units from the pole along the terminal side of the angle. Example 2
Graph Points on a Polar Grid Answer: Example 2
Graph Points on a Polar Grid Answer: Example 2
B. Graph Q(–2, –240°) on a polar grid. Graph Points on a Polar Grid B. Graph Q(–2, –240°) on a polar grid. Because = –240o, sketch the terminal side of a –240o angle with the polar axis as its initial side. Because r is negative, extend the terminal side of the angle in the opposite direction and plot a point 2 units from the pole along this extended ray. Example 2
Graph Points on a Polar Grid Answer: Example 2
Graph Points on a Polar Grid Answer: Example 2
Graph on a polar grid. A. B. C. D. Example 2
Graph on a polar grid. A. B. C. D. Example 2
Multiple Representations of Polar Coordinates Find four different pairs of polar coordinates that name point S if –360° < θ < 360°. Example 3
(2, 210°) = (2, 210o – 360°) Subtract 360° from . = (2, –150o) Multiple Representations of Polar Coordinates One pair of polar coordinates that name point S is (2, 210°). The other three representations are as follows. (2, 210°) = (2, 210o – 360°) Subtract 360° from . = (2, –150o) (2, 210°) = (–2, 210° – 180°) Replace r with –r and subtract. = (–2, 30°) 180° from . Example 3
= (–2, –150° – 180°) Replace r with –r and subtract Multiple Representations of Polar Coordinates (2, 210°) = (2, –150°) = (–2, –150° – 180°) Replace r with –r and subtract = (–2, –330°) 180° from . Answer: Example 3
= (–2, –150° – 180°) Replace r with –r and subtract Multiple Representations of Polar Coordinates (2, 210°) = (2, –150°) = (–2, –150° – 180°) Replace r with –r and subtract = (–2, –330°) 180° from . Answer: (2, –150°), (2, 210°), (–2, 30°), (–2, –330°) Example 3
Find four different pairs of polar coordinates that name point W if –360o < < 360o. B. (7, –60°), (7, 330°), (–7, 120°), (–7, 300°) C. (7, –30°), (7, 330°), (–7, 150°), (–7, –210°) D. (7, –150°), (7, 330°), (–7, 30°), (–7, 210°) Example 3
Find four different pairs of polar coordinates that name point W if –360o < < 360o. B. (7, –60°), (7, 330°), (–7, 120°), (–7, 300°) C. (7, –30°), (7, 330°), (–7, 150°), (–7, –210°) D. (7, –150°), (7, 330°), (–7, 30°), (–7, 210°) Example 3
A. Graph the polar equation r = 2.5. Graph Polar Equations A. Graph the polar equation r = 2.5. The solutions of r = 2.5 are ordered pairs of the form (2.5, ), where is any real number. The graph consists of all points that are 2.5 units from the pole, so the graph is a circle centered at the origin with radius 2.5. Answer: Example 4
A. Graph the polar equation r = 2.5. Graph Polar Equations A. Graph the polar equation r = 2.5. The solutions of r = 2.5 are ordered pairs of the form (2.5, ), where is any real number. The graph consists of all points that are 2.5 units from the pole, so the graph is a circle centered at the origin with radius 2.5. Answer: Example 4
B. Graph the polar equation . Graph Polar Equations B. Graph the polar equation . The solutions of are ordered pairs of the form , where r is any real number. The graph consists of all points on the line that makes an angle of with the positive polar axis. Example 4
Graph Polar Equations Answer: Example 4
Graph Polar Equations Answer: Example 4
Graph A. B. C. D. Example 4
Graph A. B. C. D. Example 4
Key Concept 5
Find the Distance Between Polar Coordinates A. AIR TRAFFIC An air traffic controller is tracking two airplanes that are flying at the same altitude. The coordinates of the planes are A(8, 60°) and B(4, 300°), where the directed distance is measured in miles. Sketch a graph of this situation. Airplane A is located 8 miles from the pole on the terminal side of the angle 60°, and airplane B is located 4 miles from the pole on the terminal side of the angle 300°, as shown. Example 5
Find the Distance Between Polar Coordinates Answer: Example 5
Find the Distance Between Polar Coordinates Answer: Example 5
Use the Polar Distance Formula. Find the Distance Between Polar Coordinates B. AIR TRAFFIC An air traffic controller is tracking two airplanes that are flying at the same altitude. The coordinates of the planes are A(8, 60°) and B(4, 300°), where the directed distance is measured in miles. How far apart are the two airplanes? Use the Polar Distance Formula. Polar Distance Formula Example 5
The planes are about 10.6 miles apart. Find the Distance Between Polar Coordinates (r2, 2) = (4, 300°) and (r1, 1) = (8, 60°) The planes are about 10.6 miles apart. Answer: Example 5
The planes are about 10.6 miles apart. Find the Distance Between Polar Coordinates (r2, 2) = (4, 300°) and (r1, 1) = (8, 60°) The planes are about 10.6 miles apart. Answer: about 10.6 miles Example 5
BOATS Two sailboats can be described by the coordinates (9, 60o) and (5, 320o), where the directed distance is measured in miles. How far apart are the boats? A. about 5.4 miles B. about 10.7 miles C. about 11.0 miles D. about 12.9 miles Example 5
BOATS Two sailboats can be described by the coordinates (9, 60o) and (5, 320o), where the directed distance is measured in miles. How far apart are the boats? A. about 5.4 miles B. about 10.7 miles C. about 11.0 miles D. about 12.9 miles Example 5
polar coordinate system pole polar axis polar coordinates polar equation polar graph Vocabulary