Proving the Pythagorean Identity (Grades 11-12)

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Proving the Pythagorean Identity

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

Let triangle ABC be the right triangle pictured below, where [math]theta = m ang BAC[/math] and [math]phi = m ang ABC[/math]. Let [math]a[/math] represent the leg opposite [math]ang BAC[/math], [math]b[/math] represent the leg opposite [math]ang ABC[/math], and [math]c[/math] represent the hypotenuse.

Right Triangle ABC v3

[math] \ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \ [/math][math] " Reason "[/math]
[math]1. Delta ABC " is a right triangle" \ \ \ \ \ \ \ \ \ \ \ [/math][math]1. "Given"[/math]
[math]2. a^2 + b^2 = c^2[/math][math]2. ""[/math]
[math]3. 1/c^2 (a^2 + b^2) = 1 [/math][math]3. "Multiplication Property of Equality"[/math]
[math]4. a^2/c^2 + b^2/c^2 = 1 [/math][math]4. "Distributive Property"[/math]
[math]5. (a/c)^2 + (b/c)^2 = 1 [/math][math]5. "Power of a Quotient Rule"[/math]
[math]6. [/math][math]6. "Definition of Sine in a right triangle" [/math]
[math]7. [/math][math]7. "Definition of Cosine in a right triangle"[/math]
[math]8. sin^2 theta + cos^2 theta = 1[/math][math]8. ""[/math]
A. 
What is the missing reason in step 2?
  1. Given
  2. Pythagorean Theorem
  3. Law of Sines
  4. Summation Property of Addition
B. 
What is the missing statement in step 6?
  1. [math]sin(phi) = a/c[/math]
  2. [math]sin(theta) = a/b[/math]
  3. [math]sin(theta) = b/c[/math]
  4. [math]sin(theta) = a/c[/math]
C. 
What is the missing statement in step 7?
  1. [math]cos(theta) = b/c[/math]
  2. [math]cos(theta) = a/c[/math]
  3. [math]cos(theta) = a/b[/math]
  4. [math]cos(phi) = b/c[/math]
D. 
What is the missing reason in step 8?
  1. Substitution Property of Equality
  2. Given
  3. Pythagorean Theorem
  4. Trigonometric Identity
2. 
For the following questions, let circle O be the unit circle centered at the origin and [math]theta[/math] be an angle in standard position. Let point T, with coordinates [math](x_0, y_0)[/math], be the intersection of the circle and the terminal arm of [math]theta[/math]. Let point P be [math](x_0, 0)[/math]. Also, let [math]alpha = mang TOP[/math], where [math]0 <= alpha <= pi/2[/math].
A. 
Which of the following gives the length of [math]bar(PO) ?[/math]
  1. [math]cos(alpha)[/math]
  2. [math]sin(theta)[/math]
  3. [math]cos(theta)[/math]
  4. [math]sin(alpha)[/math]
B. 
Which of the following gives the length of [math]bar(TP) ?[/math]
  1. [math]sin(theta)[/math]
  2. [math]cos(theta)[/math]
  3. [math]sin(alpha)[/math]
  4. [math]cos(alpha)[/math]
C. 
Which of the following is true?
  1. [math]cos(theta) = cos(alpha)[/math] for [math]0 < theta < pi/2[/math], but otherwise these two quantities are never equal.
  2. [math]cos(theta) = pm cos(alpha)[/math] for all values of [math]theta[/math].
  3. [math]cos(theta) = -cos(alpha)[/math] for all values of [math]theta[/math].
  4. [math]cos(theta)[/math] and [math]cos(alpha)[/math] are never equal.
D. 
Which of the following is true?
  1. [math]sin(theta) = pm sin(alpha)[/math] for [math]0 <= theta <=pi[/math]. Otherwise, these quantities are not equal.
  2. [math]sin(theta) = pm sin(alpha)[/math] for [math]0 <= theta <=pi/2[/math]. Otherwise, these quantities are not equal.
  3. [math]sin(theta)[/math] and [math]sin(alpha)[/math] are never equal.
  4. [math]sin(theta) = pm sin(alpha)[/math] for all values of [math]theta[/math].
E. 
For [math]0 < theta < 2pi, theta!=pi/2, pi, (3pi)/2[/math], what type of triangle is [math]Delta TPO?[/math]
  1. Right triangle.
  2. Acute triangle.
  3. It depends on the quadrant.
  4. Obtuse triangle.
F. 
Using the information in the previous questions, write a proof for the Pythagorean Identity, [math]sin^2(theta) + cos^2(theta) = 1[/math].



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