Difference between revisions of "2024 AMC 12A Problems/Problem 10"
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<math>\textbf{(A) }\frac{\alpha}{3}\qquad \textbf{(B) }\alpha - \frac{\pi}{8}\qquad \textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad \textbf{(D) }\frac{\alpha}{2}\qquad \textbf{(E) }\pi - 4\alpha\qquad</math> | <math>\textbf{(A) }\frac{\alpha}{3}\qquad \textbf{(B) }\alpha - \frac{\pi}{8}\qquad \textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad \textbf{(D) }\frac{\alpha}{2}\qquad \textbf{(E) }\pi - 4\alpha\qquad</math> | ||
| + | ==Solution 1== | ||
| + | From question, | ||
| + | <cmath>tan\alpha=\frac{3}{4}, \space tan\beta=\frac{7}{24}</cmath> | ||
| + | <cmath>tan(\alpha+\beta)= \frac{tan\alpha+tan\beta}{1-tan\alpha tan\beta}</cmath> | ||
| + | <cmath>tan(\alpha+\beta)= \frac{\frac{3}{4}+\frac{7}{24}}{1-\frac{3}{4} \cdot \frac{7}{24}}</cmath> | ||
| + | <cmath>tan(\alpha+\beta)=\frac{4}{3}</cmath> | ||
| + | <cmath>\alpha+\beta=tan^{-1}(\frac{4}{3})</cmath> | ||
| + | <cmath>\alpha+\beta=\frac{\pi}{2}-\alpha</cmath> | ||
| + | <math></math>\beta=\fbox{(C) <math>\frac{\pi}{2} -2\alpha</math>}<math></math> | ||
==See also== | ==See also== | ||
{{AMC12 box|year=2024|ab=A|num-b=9|num-a=11}} | {{AMC12 box|year=2024|ab=A|num-b=9|num-a=11}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 18:08, 8 November 2024
Problem
Let 
be the radian measure of the smallest angle in a 
right triangle. Let 
be the radian measure of the smallest angle in a 
right triangle. In terms of , what is ?
Solution 1
From question,
![\[tan\alpha=\frac{3}{4}, \space tan\beta=\frac{7}{24}\]](http://latex.artofproblemsolving.com/3/7/c/37c0129c887e266b41bc705db74322106c6d22a3.png)
![\[tan(\alpha+\beta)= \frac{tan\alpha+tan\beta}{1-tan\alpha tan\beta}\]](http://latex.artofproblemsolving.com/a/d/e/adea8f2d55bde236dfb35c4e1e2794ba2ae0f6da.png)
![\[tan(\alpha+\beta)= \frac{\frac{3}{4}+\frac{7}{24}}{1-\frac{3}{4} \cdot \frac{7}{24}}\]](http://latex.artofproblemsolving.com/2/0/a/20ac87bb8453be75cac2dc1ff053a7efdf992dea.png)
![\[tan(\alpha+\beta)=\frac{4}{3}\]](http://latex.artofproblemsolving.com/c/7/9/c790151177f48defd8e4fbae29f9a74b13c7f243.png)
![\[\alpha+\beta=tan^{-1}(\frac{4}{3})\]](http://latex.artofproblemsolving.com/b/5/c/b5c523e84b0a356d1c249bbbdac2525a5900a102.png)
![\[\alpha+\beta=\frac{\pi}{2}-\alpha\]](http://latex.artofproblemsolving.com/2/f/3/2f3db36a2888ec0c1261832bb95f80f5368e076f.png)
$$ (Error compiling LaTeX. Unknown error_msg)\beta=\fbox{(C) 
}$$ (Error compiling LaTeX. Unknown error_msg)
See also
| 2024 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
