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 "University of South Carolina High School Math Contest/1993 Exam/Problem 28"

 
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
 +
Suppose <math>\triangle ABC</math> is a triangle with 3 acute angles <math>A, B,</math> and <math>C</math>.  Then the point <math>( \cos B - \sin A, \sin B - \cos A)</math>
  
−
<center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ }  </math></center>
+
(A) can be in the 1st quadrant and can be in the 2nd quadrant only
 +
 
 +
(B) can be in the 3rd quadrant and can be in the 4th quadrant only
 +
 
 +
(C) can be in the 2nd quadrant and can be in the 3rd quadrant only
 +
 
 +
(D) can be in the 2nd quadrant only
 +
 
 +
(E) can be in any of the 4 quadrants
  
 
== Solution ==
 
== Solution ==
 +
First note that <math>\cos{A}, \cos{B}, \sin{A}, \sin{B}>0</math> and <math>A+B>90</math>. From these, we deduce that the <math>x</math> value and the <math>y</math> value of the coordinates will always be positive and thus the point can only be in the first quadrant.
  
 
== See also ==
 
== See also ==
 
* [[University of South Carolina High School Math Contest/1993 Exam]]
 
* [[University of South Carolina High School Math Contest/1993 Exam]]

Revision as of 19:23, 22 July 2006

Problem

Suppose $\triangle ABC$ is a triangle with 3 acute angles $A, B,$ and $C$. Then the point $( \cos B - \sin A, \sin B - \cos A)$

(A) can be in the 1st quadrant and can be in the 2nd quadrant only

(B) can be in the 3rd quadrant and can be in the 4th quadrant only

(C) can be in the 2nd quadrant and can be in the 3rd quadrant only

(D) can be in the 2nd quadrant only

(E) can be in any of the 4 quadrants

Solution

First note that $\cos{A}, \cos{B}, \sin{A}, \sin{B}>0$ and $A+B>90$. From these, we deduce that the $x$ value and the $y$ value of the coordinates will always be positive and thus the point can only be in the first quadrant.

See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=University_of_South_Carolina_High_School_Math_Contest/1993_Exam/Problem_28&oldid=8097"