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

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Proofs of Sine Addition & Subtraction Formulas

1. 
For the following proof of the sine addition formula, determine the missing statements and reasons in the questions below.

Isosceles Triangle Height v2

In triangle ABC, let point D be the intersection of the altitude (from vertex B to side [math]bar{AC}[/math]) and side [math]bar{AC}[/math] such that [math]AD + DC = AC[/math]. Let [math]m ang ABD = theta[/math] and [math]m ang DBC = alpha[/math]. Let [math]T[/math] denote the area of a triangle. Prove that [math]sin(theta + alpha) = sin(theta)cos(alpha) + sin(alpha)cos(theta)[/math].

[math] \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]
\mathbf{"Reason"}[/math]
[math]1. theta + alpha = m ang ABC [/math][math]1. "Angle Addition Postulate"[/math]
[math]2. T_{Delta ABC} = 1/2 sin(m ang ABC) AB \ BC[/math][math]2. "SAS formula for the area of a triangle"[/math]
[math]3. T_{Delta ABC} = 1/2 sin(theta+alpha) AB \ BC[/math][math]3. "Substitution Property of Equality"[/math]
[math]4. T_{Delta ABC} = 1/2 AB \ BC sin(theta+alpha) [/math][math]4. ""[/math]
[math]5. T_{Delta ABD} = 1/2 sin(theta) AB \ BD [/math][math]5. "SAS formula for the area of a triangle"[/math]
[math]6. T_{Delta BDC} = 1/2 sin(alpha) BD \ BC [/math][math]6. "SAS formula for the area of a triangle"[/math]
[math]7. bar{BD} " is an altitude of " Delta ABC [/math][math]7. ""[/math]
[math]8. bar{BD} _|_ bar{AC} [/math][math]8. "Definition of an altitude" [/math]
[math]9. [/math][math]9. "Definition of perpendicular lines"[/math]
[math]10. Delta ADB, \ Delta BDC " are right triangles"[/math][math]10. ""[/math]
[math]11. ( \ \ BD \ \ )/{BC} = cos(alpha) [/math][math]11. ""[/math]
[math]12. BD = BC cos(alpha) [/math][math]12. "Multiplication Property of Equality"[/math]
[math]13. ( \ \ BD \ \ )/{AB} = cos(theta) [/math][math]13. ""[/math]
[math]14. BD = AB cos(theta) [/math][math]14. "Multiplication Property of Equality"[/math]
[math]15. T_{Delta ABD} = 1/2 sin(theta) AB \ BC cos(alpha) [/math][math]15. "Substitution Property of Equality"[/math]
[math]16. T_{Delta ABD} = 1/2 AB \ BC sin(theta) cos(alpha) [/math][math]16. "Commutative Property of Multiplication"[/math]
[math]17. T_{Delta BDC} = 1/2 sin(alpha) AB cos(theta) BC[/math][math]17. "Substitution Property of Equality"[/math]
[math]18. T_{Delta BDC} = 1/2 AB \ BC sin(alpha) cos(theta)[/math][math]18. "Commutative Property of Multiplication"[/math]
[math]19. [/math][math]19. "Area of a shape is equal to the sum of the"[/math] [math]\ \ " areas of its partitioned shapes"[/math]
[math]20. 1/2 AB \ BC sin(theta+alpha) = 1/2 AB \ BC sin(theta) cos(alpha) [/math] [math] \ \ \ + 1/2 AB \ BC sin(alpha)cos(theta) [/math][math]20. ""[/math]
[math]21. 1/2 AB \ BC sin(theta+alpha) = [/math] [math] \ \ \ 1/2 AB \ BC (sin(theta) cos(alpha) + sin(alpha)cos(theta)) [/math][math]21. "Distribution Property of Equality"[/math]
[math]22. sin(theta+alpha) = sin(theta)cos(alpha) + sin(alpha)cos(theta) [/math][math]22. "Division Property of Equality"[/math]
A. 
What is the missing reason in step 4?
  1. Distributive Property of Multiplication
  2. Commutative Property of Multiplication
  3. SSA formula for area of a triangle
  4. Law of Sines
B. 
What is the missing reason in step 7?
  1. Given
  2. Right triangles have 2 sets of perpendicular sides
  3. Triangle Property
  4. SSS Property of Similarity
C. 
What is the missing statement in step 9?
  1. [math]ang ABC[/math] is a right angle
  2. Point [math]D[/math] is the midpoint of [math]bar{AC}[/math]
  3. [math]ang ABC[/math] is acute
  4. [math]ang ADB, \ ang BDC[/math] are right angles
D. 
What is the missing reason in step 10?
  1. SAS postulate of congruence
  2. Definition of right triangles
  3. HL postulate of congruence
  4. Pythagorean Theorem
E. 
What is the missing reason in step 11?
  1. Ratios of right triangles
  2. SAS Congruence Theorem
  3. Pythagorean Theorem
  4. SAS formula for area of a triangle
F. 
What is the missing reason in step 13?
  1. Ratios of right triangles
  2. SAS Congruence Theorem
  3. Reflexive Property of Congruence
  4. SAS formula for the area of a triangle
G. 
What is the missing statement in step 19?
  1. [math]1= T_{Delta ABD} + T_{Delta BDC}[/math]
  2. [math]T_{Delta ABC} = 2T_{Delta BDC}[/math]
  3. [math]T_{Delta ABC} = 2T_{Delta ABD}[/math]
  4. [math]T_{Delta ABC} = T_{Delta ABD} + T_{Delta BDC}[/math]
H. 
What is the missing reason in step 20?
  1. Sine Addition Formula
  2. Triangle Sum Property
  3. Substitution Property of Equality
  4. Areas of right triangles are congruent
I. 
What are some of the implicit restrictions in the above proof? Choose all correct answers.
  1. [math]0 < theta+alpha < pi[/math], where [math]theta,alpha[/math] must both be positive real numbers.
  2. [math]0 < theta+alpha < pi/2[/math], where [math]theta, alpha[/math] must both be positive real numbers.
  3. [math]Delta ABC[/math] cannot be a right triangle.
  4. [math]ang BAC[/math] and [math] angBCA[/math] must both be acute angles.
2. 
For [math]theta, alpha in RR[/math], prove that [math]sin(theta-alpha) = sin(theta)cos(alpha) - sin(alpha)cos(theta)[/math] given that [math]sin(theta + alpha) = sin(theta)cos(alpha) + sin(alpha)cos(theta)[/math].



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