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Type: Multiple-Choice
Category: Trigonometry
Level: Grade 11
Standards: HSF-TF.C.9
Author: nsharp1
Created: 4 years ago

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Trigonometry Question

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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.

[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]

Grade 11 Trigonometry CCSS: HSF-TF.C.9

What is the missing explanation in step 3?
  1. [math]"The sine and cosine functions are never greater than 1"[/math]
  2. [math]cos^2(theta)+ sin^2(theta)=1[/math]
  3. [math]cos(0)=1, \ sin(90°)=1[/math]
  4. [math]"The number 1 can be rewritten as a given quantity divided by itself"[/math]
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