2007 iTest Problems/Problem 17

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 $\tan{x}$ .

Solution

From the second equation, we get that $y=/arctan{\frac{1}{6}$ (Error compiling LaTeX. Unknown error_msg). Plugging this into the first equation, we get: $x+/arctan{\frac{1}{6}=\frac{\pi}{4}$ (Error compiling LaTeX. Unknown error_msg). Taking the tangent of both sides, $\tan{(x+/arctan{\frac{1}{6})}=\tan{\frac{\pi}{4}=1$ (Error compiling LaTeX. Unknown error_msg). From the tangent addition formula, we then get: $\tan{x}+\frac{1}{6}/1-\frac{1}{6}•\tan{x}=1$ $\tan{x}+\frac{1}{6}=1-\frac{1}{6}•\tan{x}$ . Rearranging and solving, we get: $\tan{x}=\box{\frac{5}{7}}

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2007 iTest Problems/Problem 17

Problem

Solution