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Difference between revisions of "2007 iTest Problems/Problem 17"

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If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.
 
If <math>x</math> and <math>y</math> are acute angles such that <math>x+y=\frac{\pi}{4}</math> and <math>\tan{y}=\frac{1}{6}</math>, find the value of <math>\tan{x}</math>.
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<math>\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad
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\text{(B) }\frac{35\sqrt{2}-6}{71}\qquad
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\text{(C) }\frac{35\sqrt{3}+12}{33}\qquad
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\text{(D) }\frac{37\sqrt{3}+24}{33}\qquad</math>
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<math>\text{(E) }1\qquad
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\text{(F) }\frac{5}{7}\qquad
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\text{(G) }\frac{3}{7}\qquad
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\text{(H) }6\qquad</math>
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<math>\text{(I) }\frac{1}{6}\qquad
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\text{(J) }\frac{1}{2}\qquad
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\text{(K) }\frac{6}{7}\qquad
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\text{(L) }\frac{4}{7}\qquad
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\text{(M) }\sqrt{3}\qquad</math>
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<math>\text{(N) }\frac{\sqrt{3}}{3}\qquad
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\text{(O) }\frac{5}{6}\qquad
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\text{(P) }\frac{2}{3}\qquad
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\text{(Q) }\frac{1}{2007}\qquad</math>
  
 
== Solution ==
 
== Solution ==
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Rearranging and solving, we get
 
Rearranging and solving, we get
  
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<math>\tan{x}=\boxed{\frac{5}{7}}</math>
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<math>\tan{x}=\boxed{\textbf{(F) } \frac{5}{7}}</math>
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==Solution 2==
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We will use complex numbers: <math>y</math> represents the complex number <math>6+i</math>, and <math>x+y=\frac{\pi}{4}</math> represents the fact <math>x\cdot y=1+i</math>, so dividing <math>1+i</math> by <math>6+i</math>, we get <math>\frac{7+5i}{56}</math>, which means <math>\tan{x}=\boxed{\textbf{(F) } \frac{5}{7}}</math>
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==See Also==
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{{iTest box|year=2007|num-b=16|num-a=18}}
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[[Category:Introductory Trigonometry Problems]]

Latest revision as of 20:30, 4 February 2023

Problem

If $x$ and $y$ are acute angles such that $x+y=\frac{\pi}{4}$ and $\tan{y}=\frac{1}{6}$, find the value of $\tan{x}$.

$\text{(A) }\frac{37\sqrt{2}-18}{71}\qquad \text{(B) }\frac{35\sqrt{2}-6}{71}\qquad \text{(C) }\frac{35\sqrt{3}+12}{33}\qquad \text{(D) }\frac{37\sqrt{3}+24}{33}\qquad$

$\text{(E) }1\qquad \text{(F) }\frac{5}{7}\qquad \text{(G) }\frac{3}{7}\qquad \text{(H) }6\qquad$

$\text{(I) }\frac{1}{6}\qquad \text{(J) }\frac{1}{2}\qquad \text{(K) }\frac{6}{7}\qquad \text{(L) }\frac{4}{7}\qquad \text{(M) }\sqrt{3}\qquad$

$\text{(N) }\frac{\sqrt{3}}{3}\qquad \text{(O) }\frac{5}{6}\qquad \text{(P) }\frac{2}{3}\qquad \text{(Q) }\frac{1}{2007}\qquad$

Solution

From the second equation, we get that $y=\arctan\frac{1}{6}$. Plugging this into the first equation, we get:

$x+\arctan\frac{1}{6}=\frac{\pi}{4}$

Taking the tangent of both sides,

$\tan(x+\arctan\frac{1}{6})=\tan\frac{\pi}{4}=1$

From the tangent addition formula, we then get:

$\frac{{\tan{x}+\frac{1}{6}}}{{1-\frac{1}{6}\tan{x}}}=1$

$\tan{x}+\frac{1}{6}=1-\frac{1}{6}\tan{x}$.

Rearranging and solving, we get

$\tan{x}=\boxed{\textbf{(F) } \frac{5}{7}}$

Solution 2

We will use complex numbers: $y$ represents the complex number $6+i$, and $x+y=\frac{\pi}{4}$ represents the fact $x\cdot y=1+i$, so dividing $1+i$ by $6+i$, we get $\frac{7+5i}{56}$, which means $\tan{x}=\boxed{\textbf{(F) } \frac{5}{7}}$

See Also

2007 iTest (Problems, Answer Key)
Preceded by:
Problem 16 		Followed by:
Problem 18
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