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Barnett/Ziegler/Byleen Precalculus: A Graphing Approach

Barnett/Ziegler/Byleen Precalculus: A Graphing Approach. Chapter Seven Additional Topics in Trigonometry. Law of Sines. 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). sin. sin. sin. .

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Barnett/Ziegler/Byleen Precalculus: A Graphing Approach

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  1. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Seven Additional Topics in Trigonometry

  2. Law of Sines 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). sin sin sin    = = a b c 7-1-71

  3. SSA Variations a Number of  (h = b sin ) triangles Figure Acute 0 < a < h 0 Acute a = h 1 Acute h < a < b 2 7-1-72(a)

  4. SSA Variations a Number of  (h = b sin ) triangles Figure Acute ab 1 Obtuse 0 < ab 0 Obtuse a > b 1 7-1-72(b)

  5. Law of Cosines The SAS and SSS cases are most readily solved by starting with the law of cosines. 7-2-73

  6. Vector Addition The sum of two vectors u and v can be defined using the tail-to-tiprule or the parallelogram rule: Tail-to-tip Rule Parallelogram Rule 7-3-74

  7. Algebraic Properties of Vectors 7-4-75

  8. Polar Graphing Grid 7-5-76

  9. Polar–Rectangular Relationships y P ( x , y ) 2 2 2 = + r x y P ( r ,  ) y sin = or = sin  y r  r r x y cos = or = cos  x r  r y tan =  x  x x 0 7-5-77

  10. Standard Polar Graphs—I a Line through origin: Vertical line: Horizontal line: = ar = a/cos  = a sec r = a/sin  = a cos  (a) (b) (c) 7-5-78(a)

  11. Standard Polar Graphs—I Circle: Circle: Circle:r = ar = a cos r = a sin  (d) (e) (f) 7-5-78(b)

  12. Standard Polar Graphs—II Cardioid: Cardioid: Three-leaf rose:r = a + a cos r = a + a sin r = a cos 3 (g) (h) (i) 7-5-79(a)

  13. Standard Polar Graphs—II Four-leaf rose: Lemniscate: Archimedes' spiral:r = a cos 2r2 = a2 cos 2r = aa > 0 (j) (k) (l) 7-5-79(b)

  14. Complex Numbers in Rectangular and Polar Forms z = x + iy = r(cos  + i sin  ) = rei  7-6-80

  15. De Moivre’s Theorem If z = x + iy = rei, and n is a natural number, then zn = (x + iy)n = (rei)n =rneni nth-Root Theorem For n a positive integer greater than 1, r1/ne( /n + k 360°/n)ik = 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: 7-7-81

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