Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 12B Problems/Problem 14"

(→‎Solution)
(→‎Solution: I really hate the tangent addition formula, but it's admittedly the undoubtly best sol here)
Line 13: Line 13:
 
<math>y=x^2+2x-15</math> intersects the <math>x</math>-axis at points <math>(-5, 0)</math> and <math>(3, 0)</math>. Without loss of generality, let these points be <math>A</math> and <math>C</math> respectively. Also, the graph intersects the y-axis at point <math>B = (0, -15)</math>.
 
<math>y=x^2+2x-15</math> intersects the <math>x</math>-axis at points <math>(-5, 0)</math> and <math>(3, 0)</math>. Without loss of generality, let these points be <math>A</math> and <math>C</math> respectively. Also, the graph intersects the y-axis at point <math>B = (0, -15)</math>.
  
−
Let the origin be point <math>O = (0, 0)</math>. Note that tiangles <math>AOB</math> and <math>COB</math> are right.  
+
Let point <math>O = (0, 0)</math>. Note that triangles <math>AOB</math> and <math>BOC</math> are right.  
  
−
<cmath>\tan(\angle ABC) = \frac{\sin (\angle ABC)}{\cos (\angle ABC)} = \frac{\sin(\angle ABO + \angle OBC)}{\cos(\angle ABO + \angle OBC)}</cmath>
+
<cmath>\tan(\angle ABC) = \tan(\angle ABO + \angle OBC) = \frac{\tan(\angle ABO) + \tan(\angle OBC)}{1 - \tan(\angle ABO) \cdot \tan(\angle OBC)} = \frac{\frac15 + \frac13}{1 - \frac1{15}} = \boxed{\textbf{(E)}\ \frac{4}{7}}.
 +
 
 +
Alternatively, we can use the [[Pythagorean Theorem]] to find that using the sin and cos angle addition formulas:
 +
 
 +
</cmath>\tan(\angle ABC) = \frac{\sin (\angle ABC)}{\cos (\angle ABC)} = \frac{\sin(\angle ABO + \angle OBC)}{\cos(\angle ABO + \angle OBC)} = \frac{\sin (\angle ABO) \cdot \cos (\angle OBC) + \cos (\angle ABO) \cdot \sin(\angle OBC)}{<math></math>
 +
 
 +
Alternatively, the
  
 
== See Also ==
 
== See Also ==
 
{{AMC12 box|year=2022|ab=B|num-b=13|num-a=15}}
 
{{AMC12 box|year=2022|ab=B|num-b=13|num-a=15}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 18:07, 17 November 2022

Problem

The graph of $y=x^2+2x-15$ intersects the $x$-axis at points $A$ and $C$ and the $y$-axis at point $B$. What is $\tan(\angle ABC)$?

$\textbf{(A)}\ \frac{1}{7} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{3}{7} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{4}{7} \qquad$

Solution

$y=x^2+2x-15$ intersects the $x$-axis at points $(-5, 0)$ and $(3, 0)$. Without loss of generality, let these points be $A$ and $C$ respectively. Also, the graph intersects the y-axis at point $B = (0, -15)$.

Let point $O = (0, 0)$. Note that triangles $AOB$ and $BOC$ are right. 

\[\tan(\angle ABC) = \tan(\angle ABO + \angle OBC) = \frac{\tan(\angle ABO) + \tan(\angle OBC)}{1 - \tan(\angle ABO) \cdot \tan(\angle OBC)} = \frac{\frac15 + \frac13}{1 - \frac1{15}} = \boxed{\textbf{(E)}\ \frac{4}{7}}.

Alternatively, we can use the [[Pythagorean Theorem]] to find that using the sin and cos angle addition formulas:\] (Error compiling LaTeX. Unknown error_msg)

\tan(\angle ABC) = \frac{\sin (\angle ABC)}{\cos (\angle ABC)} = \frac{\sin(\angle ABO + \angle OBC)}{\cos(\angle ABO + \angle OBC)} = \frac{\sin (\angle ABO) \cdot \cos (\angle OBC) + \cos (\angle ABO) \cdot \sin(\angle OBC)}{$$ (Error compiling LaTeX. Unknown error_msg)

Alternatively, the

See Also

2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_12B_Problems/Problem_14&oldid=182024"