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1 Name Chapter 6 Additional Topics in Trigonometry Section 6.1 Law of Sines Objective: In this lesson you learned how to use the Law of Sines to solve oblique triangles and how to find the areas of oblique triangles. Important Vocabulary Define each term or concept. Oblique triangle I. Introduction (Pages ) To solve an oblique triangle, you need to know the measure of at least one side and any two other parts of the triangle. Describe two cases that can be solved using the Law of Sines. How to use the Law of Sines to solve oblique triangles (AAS or ASA) State the Law of Sines. Example 1: For the triangle shown at the right, A = 31.6, C = 42.9, and a = 10.4 meters. Find the length of side c. A C 10.4 m B II. The Ambiguous Case (SSA) (Pages ) If two sides and one opposite angle of an oblique triangle are given, possible situations can occur, which are: How to use the Law of Sines to solve oblique triangles (SSA) 111

2 112 Chapter 6 Additional Topics in Trigonometry Example 2: For a triangle having A = 25, b = 54 feet, and a = 26 feet, how many solutions are possible? Example 3: For the triangle shown at the right, A = 110, c = 16 centimeters, and a = 25 centimeters. Find the length of side b. A 16 cm C 25 cm B III. Area of an Oblique Triangle (Page 434) The area of any triangle is the product of the lengths of two sides times the sine of. That is, Area = How to find the areas of oblique triangles Example 4: Find the area of a triangle having two sides of lengths 30 feet and 48 feet and an included angle of 40. IV. Applications of the Law of Sines (Page 435) Describe a real-life situation in which the Law of Sines could be used. How to use the Law of Sines to model and solve real-life problems Homework Assignment Page(s) Exercises

3 Section 6.2 Law of Cosines 113 Name Section 6.2 Law of Cosines Objective: In this lesson you learned how to use the Law of Cosines to solve oblique triangles and to use Heron s Formula to find the area of a triangle. I. Introduction (Pages ) State the Law of Cosines. How to use the Law of Cosines to solve oblique triangles (SSS or SAS) Example 1: Using the triangle shown at the right, find angle A. A 18 ft 13 ft When given the lengths of all three sides of a triangle and asked to find all three angles, which angle should be found first? Why? C 20 ft B Example 2: In the triangle shown at the right, if A = 62, find the length of side a. 26 ft A 19 ft C B II. Applications of the Law of Cosines (Page 441) Describe a real-life situation in which the Law of Cosines could be used. How to use the Law of Cosines to model and solve real-life problems

4 114 Chapter 6 Additional Topics in Trigonometry III. Heron s Area Formula (Page 442) Heron s Area Formula states that given any triangle with sides of length a, b, and c, the area of the triangle is: How to use Heron s Area Formula to find the area of a triangle Area = where s =. Example 3: Find the area of a triangle having sides of length a = 14 cm, b = 21 cm, and c = 27 cm. Additional notes y y y x x x Homework Assignment Page(s) Exercises

5 Section 6.3 Vectors in the Plane 115 Name Section 6.3 Vectors in the Plane Objective: In this lesson you learned how to write the component forms of vectors, perform basic vector operations, and find the direction angles of vectors. Important Vocabulary Define each term or concept. Vector v in the plane Standard position Zero vector Unit vector Standard unit vectors Direction angle I. Introduction (Page 447) A directed line segment has an and a. The magnitude of the directed line segment PQ, denoted by, is its. The magnitude of a directed line segment can be found by... How to represent vectors as directed line segments II. Component Form of a Vector (Page 448) A vector whose initial point is at the origin (0, 0) can be uniquely represented by the coordinates of its terminal point (v 1, v 2 ). This is the, written v = v 1, v 2, where v 1 and v 2 are the of v. How to write the component forms of vectors The component form of the vector with initial point P = (p 1, p 2 ) and terminal point Q = (q 1, q 2 ) is PQ = = = v.

6 116 Chapter 6 Additional Topics in Trigonometry The magnitude (or length) of v is: v = = Example 1: Find the component form and magnitude of the vector v that has (1, 7) as its initial point and (4, 3) as its terminal point. III. Vector Operations (Pages ) In operations with vectors, numbers are usually referred to as. Geometrically, the product of a vector v and a scalar k is... How to perform basic vector operations and represent them graphically If k is positive, kv has the k is negative, kv has the direction as v, and if direction. To add two vectors geometrically,... This technique is called the vector addition because the vector u + v, often called the of vector addition, is... for Let u = u 1, u 2 and v = v 1, v 2 be vectors and let k be a scalar (a real number). Then the sum of u and v is the vector: u + v = and the scalar multiple of k times u is the vector: ku = Example 2: Let u = 1, 6 and v = 4, 2. Find: (a) 3u (b) u + v

7 Section 6.3 Vectors in the Plane 117 Name Let u, v, and w be vectors and c and d be scalars. Complete the following properties of vector addition and scalar multiplication: 1. u + v = 2. (u + v) + w = 3. u + 0 = 4. u + ( u) = 5. c(du) = 6. (c + d)u = 7. c(u + v) = 8. 1(u) = 9. 0(u) = 10. cv = IV. Unit Vectors (Pages ) To find a unit vector u that has the same direction as a given nonzero vector v,... How to write vectors as linear combinations of unit vectors In this case, the vector u is called a. Example 3: Find a unit vector in the direction of v = 8, 6. Let v = v 1, v 2. Then the standard unit vectors can be used to represent v as v =, where the scalar v 1 is called the and the scalar v 2 is called the. The vector sum v 1 i + v 2 j is called a of the vectors i and j. Example 4: Let v = 5, 3. Write v as a linear combination of the standard unit vectors i and j. Example 5: Let v = 3i 4j and w = 2i + 9j. Find v + w.

8 118 Chapter 6 Additional Topics in Trigonometry V. Direction Angles (Page 453) If u is a unit vector and θ is its direction angle, the terminal point of u lies on the unit circle and u = x, y = = How to find the direction angles of vectors Now, if v is any vector that makes an angle θ with the positive x-axis, it has the same direction as u and v = = If v can be written as v = ai + bj, then the direction angle θ for v can be determined from tan θ =. Example 6: Let v = 4i + 5j. Find the direction angle for v. VI. Applications of Vectors (Pages ) Describe several real-life applications of vectors. How to use vectors to model and solve real-life problems Homework Assignment Page(s) Exercises

9 Section 6.4 Vectors and Dot Products 119 Name Section 6.4 Vectors and Dot Products Objective: In this lesson you learned how to find the dot product of two vectors and find the angle between two vectors. Important Vocabulary Define each term or concept. Angle between two nonzero vectors aaa aaaaa aa a a a a aa aaaaaaa aaa aaa aaaaaaa aaaaaaaa aaaaaaaaaa aaaaaaaa aaaaaaaa aaaaaaaa Orthogonal aaaaaaa aa aaaaa aaaaaaa aaaaaaaaaaa aaa aaaa aaaaaaa aa aaaaaaaaaaaaaaaa I. The Dot Product of Two Vectors (Pages ) The dot product of u = u 1, u 2 and v = v 1, v 2 is a a a a a a a a a a a a a. This product yields a aaaaaa. How to find the dot product of two vectors and use the Properties of the Dot Product Let u, v, and w be vectors in the plane or in space and let c be a scalar. Complete the following properties of the dot product: 1. u v = a a a a 2. 0 v = a a 3. u (v + w) = a a a a a a a a 4. v v = a 5. c(u v) = aa a a = a a aa a Example 1: Find the dot product: 5, 4 9, 2. aa II. The Angle Between Two Vectors (Pages ) If θ is the angle between two nonzero vectors u and v, then θ can be determined from aaa a a aa a aaaaaa aaaaaa. Example 2: Find the angle between v = 5, 4 and w = 9, 2. How to find the angle between two vectors and how to determine whether two vectors are orthogonal aaaaaa

10 120 Chapter 6 Additional Topics in Trigonometry An alternative way to calculate the dot product between two vectors u and v, given the angle θ between them, is a a a a aaaaa aaaaa aaa a. Two vectors u and v are orthogonal if a a a a a. Example 3: Are the vectors u = 1, 4 and v = 6, 2 orthogonal? aa III. Finding Vector Components (Pages ) Let u and v be nonzero vectors such that u = w 1 + w 2, where w 1 and w 2 are orthogonal and w 1 is parallel to (or a scalar multiple of) v. The vectors w 1 and w 2 are called aa a denoted by aaaaaa aaaaaaaaaa. The vector w 1 is the projection of u onto v and is a a a a a aa. a a aaaa a a a. The vector w 2 is given by How to write a vector as the sum of two vector components Let u and v be nonzero vectors. The projection of u onto v is given by proj v u = aaa a aaaaaaaaaa a. IV. Work (Page 466) The work W done by a constant force F as its point of application moves along the vector PQ is given by either of the following: 1. a a aaaaaa aaa aaaaaa aa 2. a a a a aa How to use vectors to find the work done by a force Homework Assignment Page(s) Exercises

11 Section 6.5 Trigonometric Form of a Complex Number 121 Name Section 6.5 Trigonometric Form of a Complex Number Objective: In this lesson you learned how to multiply and divide complex numbers written in trigonometric form and how to find powers and nth roots of complex numbers. Important Vocabulary Define each term or concept. Real axis aaa aaaaaaaaaa aaaa aa aaa aaaaaaa aaaaaa Imaginary axis aaa aaaaaaaa aaaa aa aaa aaaaaaa aaaaaa Absolute value of a complex number a + bi aaa aaaaaaaa aaaaaaa aaa aaaaaa aaa aa aaa aaa aaaaa aaa aaaa nth roots of unity aaa a aaaaaaaa aaa aaaaa aa aa I. The Complex Plane (Page 470) The complex plane is... a aaaaaaaaaa aaaaaa aa aaaaa aaaaa aaaaa aaaaaaaaaaa aa a aaaaaaa aaaaaa a a aaa On the complex plane shown at the right, (a) label the real axis, (b) label the imaginary axis, and (c) plot and label the complex numbers 2 3i and 4 + i. How to plot complex numbers in the complex plane and find absolute values of complex numbers aaaaaaaaa aaaaa 5 3 The absolute value of the complex number z = a + bi is given by a a a + bi = a a a a. II. Trigonometric Form of a Complex Number (Pages ) The trigonometric form of the complex number z = a + bi is z = aaaaa a a a aaa aa, where a = a aaa a, b = a aaa a, r = a a a a a a, and tan θ = aaa a a a -3-5 How to write the trigonometric forms of complex numbers aaaa aaaaa The number r is the aaaaaaa of z, and θ is called an aaaaaaaa of z.

12 122 Chapter 6 Additional Topics in Trigonometry The trigonometric form of a complex number is also called the aaaaa aaaa. III. Multiplication and Division of Complex Numbers (Pages ) Let z 1 = r 1 (cos θ 1 + i sin θ 1 ) and z 2 = r 2 (cos θ 2 + i sin θ 2 ) be complex numbers. Then: z 1 z 2 = a a aaaaaa a a a a a aaaaa a a aa a a a a a a How to multiply and divide complex numbers written in trigonometric form z 1 /z 2 = a aa aaaaaa a a a a a aaaaa a a aaa a a a a a a a a a a Describe how to find the product of two complex numbers. aaaaaaaa aaaaaa aaa aaa aaaaaaaaaa Describe how to find the quotient of two complex numbers. aaaaaa aaaaaa aaa aaaaaaaa aaaaaaaaaa IV. Powers of Complex Numbers (Page 474) State DeMoivre s Theorem. aa a a aaaaa a a a aaa aa aa a aaaaaaa aaaaaa aaa a aa a aaaaaaaa a a a aaaaaaaa aaaa a a aaaaaa a a a aaa aaa a a aaaa aa a a aaa aaaa How to use DeMoivre s Theorem to find powers of complex numbers IV. Roots of Complex Numbers (Pages ) The complex number u = a + bi is an nth root of the complex number z if. How to find nth roots of complex numbers For a positive integer n, the complex number z = r(cos θ + i sin θ) has aaaaaaa a aaaaaaaa aaa aaaaa given by n θ + 2πk θ + 2πk r cos + i sin, where k = 0, 1, 2,..., n 1. n n Homework Assignment Page(s) Exercises

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