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].
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.
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?
- Interior Angle Identity
- Double Angle Identity
- Given
- Law of Cosines
B.
What is the missing statement in step 5?
- [math]bar{OA}, bar{OB} " are radii of circle O"[/math]
- [math]bar{OA} " is a radius of circle O"[/math]
- [math]bar{AB} " is a radius of circle O"[/math]
- [math]bar{OB} " is a radius of circle O"[/math]
C.
What is the missing reason in step 7?
- Substitution Property of Equality
- Addition Property of Equality
- Identity Property
- Pythagorean Theorem
D.
What is the missing reason in step 10?
- Trigonometric Identity
- Length of a line segment
- Pythagorean Identity
- Law of Sines
E.
What is the missing statement in step 13?
- [math]AB^2 = cos^2(alpha)sin^2(alpha) + cos^2(theta) sin^2(theta) - 2cos(alpha)sin(alpha)cos(theta)sin(theta)[/math]
- [math]AB^2 = cos^2(alpha) - 2cos(alpha)sin(alpha) + sin^2(alpha) + cos^2(theta) - 2cos(theta)sin(theta) + sin^2(theta)[/math]
- [math]AB^2 = cos^2(alpha) - 2cos(alpha)cos(theta) + cos^2(theta) + sin^2(alpha) - 2sin(alpha)sin(theta) + sin^2(theta)[/math]
- [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?
- Addition Property of Equality
- Pythagorean Identity
- Substitution Property of Equality
- Given
G.
What is the missing reason in step 18?
- Reciprocal Identity
- Radii of the same circle are equal
- Transitive Property of Equality
- 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]
- [math]cos(theta)[/math]
- [math]-cos(theta)[/math]
- [math]-cos(-theta)[/math]
- [math]cos(theta-1)[/math]
B.
Which of the following is equal to [math]sin(-theta) ?[/math]
- [math]sin(theta)[/math]
- [math]-sin(theta)[/math]
- [math]-sin(-theta)[/math]
- [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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