1. Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Seven Additional Topics in Trigonometry

2. Copyright © 2000 by the McGraw-Hill Companies, Inc. sin  a =  b =  c The law of sines is used to solve triangles, given: 1.Two angles and any side (ASA or AAS), or 2.Two sides and an angle opposite one of them (SSA). Law of Sines 7-1-71

3. Copyright © 2000 by the McGraw-Hill Companies, Inc. aNumber of  (h = b sin  )trianglesFigure Acute 0 < a < h0 Acute a = h1 Acute h < a < b2 SSA Variations 7-1-72(a)

4. Copyright © 2000 by the McGraw-Hill Companies, Inc. aNumber of  (h = b sin  )trianglesFigure Acute a  b1 Obtuse 0 < a  b0 Obtuse a > b1 SSA Variations 7-1-72(b)

5. Copyright © 2000 by the McGraw-Hill Companies, Inc. The SAS and SSS cases are most readily solved by starting with the law of cosines. Law of Cosines 7-2-73

6. Copyright © 2000 by the McGraw-Hill Companies, Inc. Tail-to-tip Rule Parallelogram Rule Vector Addition The sum of two vectors u and v can be defined using the tail-to-tip rule or the parallelogram rule: 7-3-74

7. Copyright © 2000 by the McGraw-Hill Companies, Inc. Algebraic Properties of Vectors 7-4-75

8. Copyright © 2000 by the McGraw-Hill Companies, Inc. Polar Graphing Grid 7-5-76

9. Copyright © 2000 by the McGraw-Hill Companies, Inc. r 2 = x 2 + y 2 sin  = y r or y = r sin  cos  = x r or x = r cos  tan  = y x x x y y r  P(x,y) P(r,  ) 0 Polar–Rectangular Relationships 7-5-77

10. Copyright © 2000 by the McGraw-Hill Companies, Inc. Standard Polar Graphs—I Line through origin: Vertical line: Horizontal line:  = a r = a/cos  = a sec  r = a/sin  = a cos  (a) (b) (c) 7-5-78(a) a

11. Copyright © 2000 by the McGraw-Hill Companies, Inc. Standard Polar Graphs—I Circle: Circle: Circle: r = a r = a cos  r = a sin  (d) (e)(f) 7-5-78(b)

12. Copyright © 2000 by the McGraw-Hill Companies, Inc. Cardioid: Cardioid: Three-leaf rose: r = a + a cos  r = a + a sin  r = a cos 3  (g)(h) (i) Standard Polar Graphs—II 7-5-79(a)

13. Copyright © 2000 by the McGraw-Hill Companies, Inc. Four-leaf rose: Lemniscate: Archimedes' spiral: r = a cos 2  r 2 = a 2 cos 2  r = a  a > 0 (j) (k) (l) Standard Polar Graphs—II 7-5-79(b)

14. Copyright © 2000 by the McGraw-Hill Companies, Inc. Complex Numbers in Rectangular and Polar Forms z= x + iy = r(cos  + i sin  ) = re i  7-6-80 

15. Copyright © 2000 by the McGraw-Hill Companies, Inc. For n a positive integer greater than 1, r 1/n e (  /n + k 360°/n)i k = 0, 1, …, n – 1 are the n distinct nth roots of re  i and there are no others. The four distinct fourth roots of –1 are: De Moivre’s Theorem If z = x + iy = re i , and n is a natural number, then z n = (x + iy) n = (re i  ) n = r n e n  i 7-7-81 nth-Root Theorem