Proofs of Cosine Addition & Subtraction Formulas (Grades 11-12)

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Proofs of Cosine Addition & Subtraction Formulas

1. 
For the following proof, determine the missing statements and reasons in the questions below.

Let circle O be the unit circle, whose center is the origin. Let [math]theta[/math] and [math]alpha[/math] be two angles in standard position, such that [math]0 < alpha < theta < 2pi[/math]. Let point [math]A[/math] be the intersection of the terminal arm of angle [math]theta[/math] and circle O, with coordinates [math](x_A, y_A)[/math]. Let point [math]B[/math] be the intersection of the terminal arm of angle [math]alpha[/math] and circle O, with coordinates [math](x_B, y_B)[/math]. Prove that [math]cos(theta-alpha) = cos(theta)cos(alpha) + sin(theta)sin(alpha)[/math].

[math] \ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] " Reason "[/math]
[math]1. m ang AOB + alpha = theta \ \ \ \ \ \ \ \ \ \ \ [/math][math]1. "Angle Addition Postulate"[/math]
[math]2. m ang AOB = theta - alpha[/math][math]2. "Subtraction Property of Equality"[/math]
[math]3. AB^2 = OA^2 + OB^2 - 2OA \ OBcos(m ang AOB) [/math][math]3. ""[/math]
[math]4. AB^2 = OA^2 + OB^2 - 2OA \ OBcos(theta-alpha) [/math][math]4. "Substitution Property of Equality"[/math]
[math]5. [/math][math]5. "Definition of radius"[/math]
[math]6. OA = OB = 1 [/math][math]6. "Given (circle O is the unit circle)"[/math]
[math]7. AB^2 = 1+1-2(1)(1)Cos(theta-alpha) [/math][math]7. ""[/math]
[math]8. AB^2 = 2-2cos(theta-alpha) [/math][math]8. "Simplification (add, multiply)" [/math]
[math]9. (x_A, y_A) = (cos(theta), sin(theta))[/math], [math] \ \ \ (x_B, y_B) = (cos(alpha), sin(alpha))[/math][math]9. "For any angle " gamma " in standard position,"[/math] [math] \ \ \ "the coordinates of the intersection"[/math] [math] \ \ \ "of its terminal arm and the unit circle"[/math] [math] \ \ \ "are " (cos(gamma), sin(gamma))[/math]
[math]10. AB = sqrt((x_B - x_A)^2 + (y_B - y_A)^2) [/math][math]10. ""[/math]
[math]11. AB^2 = (x_B - x_A)^2 + (y_B - y_A)^2 [/math][math]11. "Multiplication Property of Equality"[/math]
[math]12. AB^2 = (cos(alpha)-cos(theta))^2 + (sin(alpha)-sin(theta))^2 [/math][math]12. "Substitution Property of Equality"[/math]
[math]13. [/math][math]13. "Distributive Property of Equality"[/math]
[math]14. AB^2 = cos^2(theta) + sin^2(theta) + cos^2(alpha) + sin^2(alpha) [/math] [math] \ \ \ - 2cos(theta)cos(alpha) - 2sin(theta)sin(alpha) [/math][math]14. "Commutative Property of Addition,"[/math] [math] \ \ \ "Commutative Property of Multiplication"[/math]
[math]15. AB^2 = 1 + 1 - 2cos(theta)cos(alpha) - 2sin(theta)sin(alpha) [/math][math]15. ""[/math]
[math]16. AB^2 = 1 + 1 - 2[ \ cos(theta)cos(alpha) + sin(theta)sin(alpha) \ ] [/math][math]16. "Distributive Property of Equality"[/math]
[math]17. AB^2 = 2 - 2[ \ cos(theta)cos(alpha) + sin(theta)sin(alpha) \ ] [/math][math]17. "Simplification (add)"[/math]
[math]18. 2 - 2cos(theta-alpha) = 2 - 2[ \ cos(theta)cos(alpha) + sin(theta)sin(alpha) \ ] [/math][math]18. ""[/math]
[math]19. - 2cos(theta-alpha) = - 2[ \ cos(theta)cos(alpha) + sin(theta)sin(alpha) \ ] [/math][math]19. "Subtraction Property of Equality"[/math]
[math]20. cos(theta-alpha) = cos(theta)cos(alpha) + sin(theta)sin(alpha) [/math][math]20. "Division Property of Equality"[/math]


Note: If [math]theta - alpha= m ang AOB > pi[/math], then the equation in step 3 will be [math]AB^2 = OA^2 + OB^2 - 2OA \ OB cos(2pi - m ang AOB)[/math]. However, since [math]cos(2pi - x) = cos(x), forall x in RR[/math], then [math]cos(2pi - m ang AOB) = cos(m ang AOB)[/math], and the proof remains the same.
Also, if [math]theta - alpha = m ang AOB = pi[/math], this proof is not valid. However, the equation to be proved simply becomes, on the LHS, [math]-1[/math], and on the RHS [math]cos(theta)cos(alpha)+sin(theta)sin(alpha) = [/math] [math]cos(pi + alpha)cos(alpha) + sin(pi + alpha)sin(alpha) = [/math] [math]-cos^2(alpha) - sin^2(alpha) = -1[/math], proving the equation is true for this special case as well.
A. 
What is the missing reason in step 3?
  1. Interior Angle Identity
  2. Double Angle Identity
  3. Given
  4. Law of Cosines
B. 
What is the missing statement in step 5?
  1. [math]bar{OA}, bar{OB} " are radii of circle O"[/math]
  2. [math]bar{OA} " is a radius of circle O"[/math]
  3. [math]bar{AB} " is a radius of circle O"[/math]
  4. [math]bar{OB} " is a radius of circle O"[/math]
C. 
What is the missing reason in step 7?
  1. Substitution Property of Equality
  2. Addition Property of Equality
  3. Identity Property
  4. Pythagorean Theorem
D. 
What is the missing reason in step 10?
  1. Trigonometric Identity
  2. Length of a line segment
  3. Pythagorean Identity
  4. Law of Sines
E. 
What is the missing statement in step 13?
  1. [math]AB^2 = cos^2(alpha)sin^2(alpha) + cos^2(theta) sin^2(theta) - 2cos(alpha)sin(alpha)cos(theta)sin(theta)[/math]
  2. [math]AB^2 = cos^2(alpha) - 2cos(alpha)sin(alpha) + sin^2(alpha) + cos^2(theta) - 2cos(theta)sin(theta) + sin^2(theta)[/math]
  3. [math]AB^2 = cos^2(alpha) - 2cos(alpha)cos(theta) + cos^2(theta) + sin^2(alpha) - 2sin(alpha)sin(theta) + sin^2(theta)[/math]
  4. [math]AB^2 = cos^2(alpha) + cos^2(theta) + sin^2(alpha) + sin^2(theta)[/math]
F. 
What is the missing reason in step 15?
  1. Addition Property of Equality
  2. Pythagorean Identity
  3. Substitution Property of Equality
  4. Given
G. 
What is the missing reason in step 18?
  1. Reciprocal Identity
  2. Radii of the same circle are equal
  3. Transitive Property of Equality
  4. Pythagorean Theorem
2. 
For the following questions, let [math]theta, alpha in RR[/math].
A. 
Which of the following is equal to [math]cos(-theta) ?[/math]
  1. [math]cos(theta)[/math]
  2. [math]-cos(theta)[/math]
  3. [math]-cos(-theta)[/math]
  4. [math]cos(theta-1)[/math]
B. 
Which of the following is equal to [math]sin(-theta) ?[/math]
  1. [math]sin(theta)[/math]
  2. [math]-sin(theta)[/math]
  3. [math]-sin(-theta)[/math]
  4. [math]sin(theta-1)[/math]
C. 
Given the information in the previous two questions, along with the fact that [math]cos(theta - alpha) = cos(theta)cos(alpha) + sin(theta)sin(alpha)[/math], prove that [math]cos(theta + alpha) = cos(theta)cos(alpha) - sin(theta)sin(alpha)[/math].






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