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 "1982 AHSME Problems/Problem 23"
m
Line 14: Line 14:
 
We apply the Law of Cosines to solve for <math>\cos\angle A:</math> <cmath>\cos\theta=\frac{b^2+c^2-a^2}{2bc}=\frac{n+5}{2(n+2)}.</cmath>
 
We apply the Law of Cosines to solve for <math>\cos\angle A:</math> <cmath>\cos\theta=\frac{b^2+c^2-a^2}{2bc}=\frac{n+5}{2(n+2)}.</cmath>
  
−
Let the brackets denote areas. We find <math>[ABC]</math> using <math>\sin\angle A</math> and <math>\sin\angle C:</math> <cmath>[ABC]=\frac12 bc\sin\theta=\frac12 ab\sin(2\theta).</cmath>
+
Let the brackets denote areas. We write <math>[ABC]</math> in terms of <math>\sin\angle A</math> and <math>\sin\angle C,</math> respectively: <cmath>[ABC]=\frac12 bc\sin\theta=\frac12 ab\sin(2\theta).</cmath>
 
Recall that <math>\sin(2\theta)=2\sin\theta\cos\theta</math> holds for all <math>\theta.</math> Equating the last two expressions and then simplifying, we have <cmath>\cos\theta=\frac{c}{2a}=\frac{n+2}{2n}.</cmath>
 
Recall that <math>\sin(2\theta)=2\sin\theta\cos\theta</math> holds for all <math>\theta.</math> Equating the last two expressions and then simplifying, we have <cmath>\cos\theta=\frac{c}{2a}=\frac{n+2}{2n}.</cmath>
 
Equating the expressions for <math>\cos\theta,</math> we get <cmath>\frac{n+5}{2(n+2)}=\frac{n+2}{2n},</cmath>
 
Equating the expressions for <math>\cos\theta,</math> we get <cmath>\frac{n+5}{2(n+2)}=\frac{n+2}{2n},</cmath>
Line 22: Line 22:
  
 
== Solution 2 ==
 
== Solution 2 ==
−
In <math>\triangle ABC,</math> let <math>a=n,b=n+1,c=n+2,</math> and <math>\angle A=\theta</math> for some positive integer <math>n.</math> We are given that <math>\angle C=2\theta,</math> and we need <math>\cos\theta.</math>
+
This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that <math>\cos\theta=\frac{n+5}{2(n+2)}</math> from the second paragraph of Solution 1.
 +
 
 +
Let the brackets denote areas. We write <math>[ABC]</math> using <math>\sin\angle A</math> and Heron's Formula, respectively:
 +
 
 +
~MRENTHUSIASM
 +
 
 +
== Solution 3 ==
 +
This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that <math>\cos\theta=\frac{n+5}{2(n+2)}</math> from the second paragraph of Solution 1.
  
−
We apply the Law of Cosines to solve for <math>\cos\angle A:</math> <cmath>\cos\theta=\frac{b^2+c^2-a^2}{2bc}=\frac{n+5}{2(n+2)}.</cmath>
 
 
We apply the Law of Cosines to solve for <math>\cos\angle C:</math> <cmath>\cos(2\theta)=\frac{a^2+b^2-c^2}{2ab}=\frac{n-3}{2n}.</cmath>
 
We apply the Law of Cosines to solve for <math>\cos\angle C:</math> <cmath>\cos(2\theta)=\frac{a^2+b^2-c^2}{2ab}=\frac{n-3}{2n}.</cmath>
 
By the Double-Angle Formula <math>\cos(2\theta)=2\cos^2\theta-1,</math> we have <cmath>2\left(\frac{n+5}{2(n+2)}\right)^2-1=\frac{n-3}{2n},</cmath> from which <math>n=-3,-\frac12,4.</math> Recall that <math>n</math> is a positive integer, so <math>n=4.</math> By substitution, the answer is <math>\cos\theta=\boxed{\textbf{(A)}\ \frac{3}{4}}.</math>
 
By the Double-Angle Formula <math>\cos(2\theta)=2\cos^2\theta-1,</math> we have <cmath>2\left(\frac{n+5}{2(n+2)}\right)^2-1=\frac{n-3}{2n},</cmath> from which <math>n=-3,-\frac12,4.</math> Recall that <math>n</math> is a positive integer, so <math>n=4.</math> By substitution, the answer is <math>\cos\theta=\boxed{\textbf{(A)}\ \frac{3}{4}}.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
−
 
−
== Solution 3 ==
 
  
 
== See Also ==
 
== See Also ==
 
{{AHSME box|year=1982|num-b=22|num-a=24}}
 
{{AHSME box|year=1982|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 01:16, 15 September 2021

Problem

The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is

$\textbf{(A)}\ \frac{3}{4}\qquad \textbf{(B)}\ \frac{7}{10}\qquad \textbf{(C)}\ \frac{2}{3}\qquad \textbf{(D)}\ \frac{9}{14}\qquad \textbf{(E)}\ \text{none of these}$

Solution 1

In $\triangle ABC,$ let $a=n,b=n+1,c=n+2,$ and $\angle A=\theta$ for some positive integer $n.$ We are given that $\angle C=2\theta,$ and we need $\cos\theta.$

We apply the Law of Cosines to solve for $\cos\angle A:$ \[\cos\theta=\frac{b^2+c^2-a^2}{2bc}=\frac{n+5}{2(n+2)}.\]

Let the brackets denote areas. We write $[ABC]$ in terms of $\sin\angle A$ and $\sin\angle C,$ respectively: \[[ABC]=\frac12 bc\sin\theta=\frac12 ab\sin(2\theta).\] Recall that $\sin(2\theta)=2\sin\theta\cos\theta$ holds for all $\theta.$ Equating the last two expressions and then simplifying, we have \[\cos\theta=\frac{c}{2a}=\frac{n+2}{2n}.\] Equating the expressions for $\cos\theta,$ we get \[\frac{n+5}{2(n+2)}=\frac{n+2}{2n},\] from which $n=4.$ By substitution, the answer is $\cos\theta=\boxed{\textbf{(A)}\ \frac{3}{4}}.$

~MRENTHUSIASM

Solution 2

This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that $\cos\theta=\frac{n+5}{2(n+2)}$ from the second paragraph of Solution 1.

Let the brackets denote areas. We write $[ABC]$ using $\sin\angle A$ and Heron's Formula, respectively:

~MRENTHUSIASM

Solution 3

This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that $\cos\theta=\frac{n+5}{2(n+2)}$ from the second paragraph of Solution 1.

We apply the Law of Cosines to solve for $\cos\angle C:$ \[\cos(2\theta)=\frac{a^2+b^2-c^2}{2ab}=\frac{n-3}{2n}.\] By the Double-Angle Formula $\cos(2\theta)=2\cos^2\theta-1,$ we have \[2\left(\frac{n+5}{2(n+2)}\right)^2-1=\frac{n-3}{2n},\] from which $n=-3,-\frac12,4.$ Recall that $n$ is a positive integer, so $n=4.$ By substitution, the answer is $\cos\theta=\boxed{\textbf{(A)}\ \frac{3}{4}}.$

~MRENTHUSIASM

See Also

1982 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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 • 26 • 27 • 28 • 29 • 30
All AHSME 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=1982_AHSME_Problems/Problem_23&oldid=162273"