Chapter 6 Additional Topics in Trigonometry

Transcription

1 Chapter 6 Additional Topics in Trigonometry Overview: 6.1 Law of Sines 6.2 Law of Cosines 6.3 Vectors in the Plan 6.4 Vectors and Dot Products

2 6.1 Law of Sines What You ll Learn: #115 - Use the Law of Sines to solve oblique triangles. AAS, ASA, or SSA Triangles #116 - Find areas of oblique triangles. #117 - Use the Law of Sines to model and solve real-life problems.

3 Four Possible Cases to Solve Triangle 1. One side and two angles are known (ASA or SAA) 2. Two sides and the angle opposite one of them (SSA) 3. Two sides and the included angle are known (SAS) 4. Three sides are known (SSS)

4 The Law of Sines sin A a sin B = b = sin C c

5 Using Law of Sines to Solve a Solve the triangle: A = 40 o, B = 60 o, a = 4 B c a A b C

6 Using Law of Sines to Solve a Solve the triangle: A = 35 o, B = 15 o, c = 5 B c a A b C

7 The Ambiguous Case Given angle A, side a, and side b Two possible Triangles Only one possible triangle No triangle exists

8 Two Possible Triangles

9 Using Law of Sines to Solve a B Solve the triangle: a = 3, b = 2, A = 40 o c a A b C

10 Using Law of Sines to Solve a B Solve the triangle: a = 2, b = 1, B = 50 o c a A b C

11 Using Law of Sines to Solve a B Solve the triangle: a = 6, b = 8, A = 35 o c a A b C

12 Applications Find the distance from the Ranger s Tower to the fire. x

13 Applications Find the height of the mountain.

14 Back to the beach Find the distance out to the island from the beach. water

15 The Area of a Triangle (TRIG VERSION) sin C = h b h = b sin C A = 1 2 bh is now Area = 1 ab sin C 2

16 The Area Formulas A = 1 2 ab sin C c B A = 1 2 bc sin A A a b A = 1 2 ac sin B C

17 Find the Area of a SAS Find the area A of the triangle for which a = 8, b = 6, and C = 30 o. A C B

18 Homework Page 398 #1,2,7,15-19,21,22

19 6.2 Law of Cosines What You ll Learn: #118 - Use the Law of Cosines to solve oblique triangles. SSS or SAS Triangles #119 - Use the Law of Cosines to model and solve real-life problems. #120 - Use Heron s Area Formula to find areas of triangles.

20 The Law of Cosines c 2 = a 2 + b 2 2ab cos C b 2 = a 2 + c 2 2ac cos B a 2 = b 2 + c 2 2bc cos A

21 Four Possible Cases to Solve Triangle 1. One side and two angles are known (ASA or SAA) 2. Two sides and the angle opposite one of them (SSA) 3. Two sides and the included angle are known (SAS) 4. Three sides are known (SSS)

22 Using Law of Cosines to Solve a B Solve the triangle: a = 2, b = 3, C = 60 o c a A b C

23 Using Law of Cosines to Solve a B Solve the triangle: a = 4, b = 3, c = 6 c a A b C

24 Applications Progressive Field X 90 Feet

25 Applications Michael Brantley X 90 Feet

26 Applications A motorized sailboat leaves Naples, Florida, bound for Key West, 150 miles away. Maintaining a constant speed of 15 miles per hour, but encountering heavy crosswinds and currents, the crew finds, after 4 hours, that the sailboat is off course by 20 o. a) How far is the sailboat from Key West at this time? b) Through what angle should the sailboat turn to correct its course? c) How much time has been added to the trip because of this? (Assume speed remains 15 mph)

27 Applications Naples You are here 20 o Key West

28 Key West Highway (US RT 1)

29 Key West Highway (US RT 1)

30 Area of a SSS Triangle Heron s Formula A = s(s a)(s b)(s c) s = 1 2 (a + b + c)

31 Example Find the area SSS Sides lengths: a = 4, b = 5, c = 7 a b c

32 Homework Page 405 #1-3,19,20,27,32,35,36

33 Chapter 6 Review Page 444 #1-29 Odd, #33,35

34 6.3 Vectors in the Plane What You ll Learn: Represent vectors as directed line segments Write the component form of vectors Perform basic vector operations and represent vectors graphically Write vectors as linear combinations of unit vectors Find the direction angles of vectors Use vectors to model and solve real-life problems.

35 Vectors A vector is a quantity that has magnitude and direction. AB B A Terminal point Initial point

36 Vectors The magnitude of a vector is its length, and is denoted by AB. In the coordinate plane, use distance formula. B(3, 2) AB = y 2 y x 2 x 2 1 AB = AB = 13 A (0, 0) AB

37 Component Form of a Vector A vector in standard position has an initial point at the origin. The component form of a vector may be written as V = v 1, v 2 where v 1 is the x coordinate of the terminal point and v 2 is the y coordinate of the terminal point. Between two points P (p 1, p 2 ) and Q (q 1, q 2 ) the component form will be PQ = q 1 p 1, q 2 p 2 = v 1, v 2 Two vectors are equal if their components are equal

38 Finding Magnitude Given Component Form Since component form already found the difference of the coordinates, you may use the components to find the magnitude of a vector. v 1 = x 2 x 1 v 2 = y 2 y 1 AB = v v 2 2

39 Example Finding component form Find the component form and magnitude of vector V that has initial point (4, 7) and terminal point 1,5. V = q 1 p 1, q 2 p 2 = 1 4, 5 ( 7) = 5, 12 Magnitude = V = v v 2 2 = = 169 = 13

40 Vector Operations v u v + u u v v + 2u

41 Vector Addition and Scalar Multiples Let u = u 1, u 2 and v = v 1, v 2 The sum of u and v will be the vector u + v = u 1 + v 1, u 2 + v 2 The scalar multiple of k times u will be the vector ku = k u 1, u 2 = ku 1, ku 2

42 Vector Operations Let v = 2, 5 and w = 3, 4 and find each of the following vectors. 1. 3v 2. w v 3. v + 2w

43 Properties of Vector Addition and Scalar Multiplication Let u, v, and w be vectors and let c and d be scalar multiples. Then the following properties are true. 1. u + v = v + u 2. u + v + w = u + (v + w) 3. u + 0 = u 4. u + u = 0 5. c du = cd u 6. c + d u = cu + du 7. c u + v = cu + cv 8. 1 u = u 9. 0 u = cv = c v

44 Unit Vectors When you are working on applications involving vectors, it is often times very helpful to know the unit vector. The unit vector captures a quantity s direction with a magnitude of one unit. unit vector = v v unit vector in direction of v

45 Finding a Unit Vector Find a unit vector in the direction of v = 2, 5 and verify that the result has a magnitude of 1. Verification of the unit vector having magnitude of one unit = = = 1

46 Linear Combination Vectors may be represented with horizontal and vertical components. v = v 1, v 2 v = v 1 1,0 + v 2 0,1 1, 0 Horizontal unit vector 0, 1 vertical unit vector v = v 1 i + v 2 j

47 Writing a Linear Combination Vector Let u be the vector with initial point (2, 5) and terminal point ( 1, 3). Write u as a linear combination of the standard unit vectors i and j.

48 Vector Operations Let u = 3i + 8j and v = 2i j. Find 2u 3v.

49 Direction Angles of Vectors u = x, y u = cos θ, sin θ u = cos θ i + (sin θ)j The angle θ is the direction angle measured counterclockwise from the positive x-axis. (like unit circle)

50 Finding Direction Angles of Vectors Find the direction angle of the vector. u = 3i + 3j

51 Applications of Vectors Find the component form of the vector that represents the velocity of an airplane descending at a speed of 100 miles per hour at angle of 30 o below the horizontal. 210 o v = v cos θ i + v sin θ j

52 Applications of Vectors An airplane is traveling at a speed of 500 miles per hour with a bearing of 330 o at a fixed altitude with a negligible wind velocity. As the airplane reaches a certain point, it encounters a wind blowing with a velocity of 70 miles per hour in the direction N 45 o E. What are the resultant speed and direction of the airplane?

53 Homework Page 417 #1-11 ODD,19,23,27,31,41,42,47,50-53,69,81

54 6.4 Vectors and Dot Products What You ll Learn: Find the dot product of two vectors and use properties of the dot product Find angles between vectors and determine whether two vectors are orthogonal Write vectors as sums of two vector components Use vectors to find the work done by a force

55 Dot Product The dot product of u = u 1, u 2 v = v 1, v 2 is given by and u v = u 1 v 1 + u 2 v 2

56 Properties of the Dot Product Let u, v, and w be vectors in the plane or in space and let c be a scalar. 1. u v = v u 2. 0 v = 0 3. u v + w = u v + u w 4. v v = v 2 5. c u v = cu v = u cv

57 Finding Dot Products Find each dot product. 1. 4,5 2,3 2. 2, 1 1,2 3. 0,3 4, 2

58 Using Properties of Dot Products Let u = 1,3, v = 2, 4 and w = 1, 2. Find each. 1. u v w 2. u 2v

59 The Angle Between Two Vectors If θ is the angle between two nonzero vectors u and v, then cos θ = u v u v

60 Finding the Angle Between Vectors Find the angle between: u = 4,3 and v = 3,5 u = 2, 6 and v = 6,2

61 Homework Page 429 #1-23 ODD

62 Review Chapter 6 Page 444 #1, 2, 6, 11, 15, 19, 21, 30, 33 *6.1 & 6.2 #37-51, ALL ODD *6.3 #75-81 *6.4 #69 Static Equilibrium