Proofs of Tangent Addition & Subtraction Formulas (Grades 11-12)
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Proofs of Tangent Addition & Subtraction Formulas
1.
For the following proof of the tangent addition formula, choose the correct missing expression or explanation for each applicable step in the questions below.
For [math]alpha, beta in RR[/math], prove that [math]tan(alpha+beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))[/math]. Assume that the addition formulas for sine and cosine are known.
For [math]alpha, beta in RR[/math], prove that [math]tan(alpha+beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))[/math]. Assume that the addition formulas for sine and cosine are known.
[math] [/math] | [math] \ \ \ \mathbf{ "Algebraic Steps" } \ \ \ [/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math]\mathbf{ "Explanation" }[/math] |
[math][/math] | [math] (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta)) [/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math]"(R.H.S. of the given equation)"[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]1.[/math] | [math] \ \ \ \ \ = (sin(alpha)/cos(alpha) + sin(beta)/cos(beta))/(1-sin(alpha)/cos(alpha) sin(beta)/cos(beta))[/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]2. [/math] | [math] \ \ \ \ \ = [/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math] "Combine fractions in numerator"[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]3. [/math] | [math] \ \ \ \ \ = ((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta)))/((cos(alpha)cos(beta))/(cos(alpha)cos(beta)) - (sin(alpha)sin(beta))/(cos(alpha)cos(beta)))[/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]4. [/math] | [math] \ \ \ \ \ = ((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta)))/((cos(alpha)cos(beta) \ - \ sin(alpha)sin(beta))/(cos(alpha)cos(beta)))[/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math] "Combine fractions in the denominator"[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]5. [/math] | [math] \ \ \ \ \ = [/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math] "Division of fractions"[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]6. [/math] | [math] \ \ \ \ \ = (sin(alpha)cos(beta) + sin(beta)cos(alpha))/(cos(alpha)cos(beta) - sin(alpha)sin(beta))[/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math]""[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]7. [/math] | [math] \ \ \ \ \ = sin(alpha+beta)/(cos(alpha)cos(beta) - sin(alpha)sin(beta))[/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math]""[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]8. [/math] | [math] \ \ \ \ \ = sin(alpha+beta)/cos(alpha+beta)[/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math]""[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
[math]9. [/math] | [math] \ \ \ \ \ = tan(alpha+beta)[/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math] "Definition of tangent"[/math] |
[math][/math] | [math][/math] | [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] | [math][/math] |
A.
What is the missing explanation in step 1?
- [math]"Definition of tangent"[/math]
- [math]"Pythagorean Identity"[/math]
- [math]"Law of S""ines"[/math]
- [math]alpha/alpha = 1, \ beta/beta = 1[/math]
B.
What is the missing expression in step 2?
- [math](((sin(alpha)cos(beta))/(cos(alpha)sin(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math]
- [math](((sin(alpha)sin(beta))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math]
- [math](((sin(alpha)cos(alpha) \ + \ sin(beta)cos(beta))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math] [math] \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [/math]
- [math](((sin(alpha)cos(beta) \ + \ sin(beta)cos(alpha))/(cos(alpha)cos(beta))))/((1-sin(alpha)/cos(alpha) sin(beta)/cos(beta)))[/math]
C.
What is the missing explanation in step 3?
- [math]"The sine and cosine functions are never greater than 1"[/math]
- [math]cos^2(theta)+ sin^2(theta)=1[/math]
- [math]cos(0)=1, \ sin(90°)=1[/math]
- [math]"The number 1 can be rewritten as a given quantity divided by itself"[/math]
D.
What is the missing expression in step 5?
[math]( ( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))[/math] [math] \ \ [/math] [math]( ( \ cos(alpha)cos(beta) \ )( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/(( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))[/math] [math] \ \ [/math] [math]( ( \ cos(alpha)cos(beta) \ )( \ sin(alpha)cos(beta) + sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ cos(alpha)cos(beta) - sin(alpha)sin(beta) \ ))[/math] [math] \ \ [/math] [math]( ( \ sin(alpha)cos(beta) \ )( \ sin(beta)cos(alpha) \ ))/( ( \ cos(alpha)cos(beta) \ ) ( \ sin(alpha)sin(beta) \ ))[/math] [math] \ \ [/math]
E.
What is the missing explanation in step 6?
- Pythagorean Identity
- Cancel like factors
- Substitution Property of Equality
- Division Property of Identity
F.
What is the missing explanation in step 7?
- Substitute in the sine addition formula
- Apply the Law of Sines
- Cancel like terms
- Factor out like terms
G.
What is the missing explanation in step 8?
- Substitute in the cosine addition formula
- Apply the Law of Cosines
- Cancel like terms
- Factor out like terms
2.
Given [math]tan(alpha + beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))[/math] for [math]alpha, beta in RR[/math], prove that [math]tan(alpha-beta) = (tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta))[/math].
A.
Which of the following is equal to [math]tan(-x) ?[/math]
- [math]tan(x)[/math]
- [math]-tan(x)[/math]
- [math]-tan(-x)[/math]
- [math]tan(x-1)[/math]
B.
Prove [math]tan(alpha-beta) = (tan(alpha)-tan(beta))/(1+tan(alpha)tan(beta))[/math], given that [math]tan(alpha + beta) = (tan(alpha)+tan(beta))/(1-tan(alpha)tan(beta))[/math].
3.
Which of the following is equal to [math]tan(alpha-beta)/tan(alpha+beta) ?[/math]
- [math](sin(alpha)cos(alpha)) / (sin(beta)cos(beta))[/math]
- [math](sin(alpha)cos(beta) - sin(beta)cos(alpha)) / (sin(alpha)cos(alpha) + sin(beta)cos(beta))[/math]
- [math](sin(alpha)cos(alpha) - sin(beta)cos(beta)) / (sin(alpha)cos(alpha) + sin(beta)cos(beta))[/math]
- [math](sin(alpha)cos(alpha) - sin(beta)cos(beta)) / (sin(alpha)cos(beta) + sin(beta)cos(alpha))[/math]
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