Difference between revisions of "2006 AMC 12A Problems/Problem 15"

Revision as of 22:17, 5 February 2016

Problem

Suppose $\cos x=0$ and $\cos (x+z)=1/2$ . What is the smallest possible positive value of $z$ ?

$\mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6} \quad \mathrm{(E) \ } \frac{7\pi}{6}$

Solutions

Solution 1

For $\cos x = 0$ , x must be in the form of $\frac{\pi}{2} + \pi n$ , where $n$ denotes any integer.
For $\cos (x+z) = 1 / 2$ , $x + z = \frac{\pi}{3} +2\pi n, \frac{5\pi}{3} + 2\pi n$ .

The smallest possible value of $z$ will be that of $\frac{5\pi}{3} - \frac{3\pi}{2} = \frac{\pi}{6} \Rightarrow \mathrm{(A)}$ .

Solution 2

Since $cos(x) = 0$ , we know that x must equal some multiple of 90. Let us assume x = 90. We want $cos(x+z) = 1/2$ , and by using the property that $cos(x) = cos(180-x)$ , we want x = 60 since $cos(60) = \frac{1}{2}$ . This means that we have $x + z = 120$ , and from this we see that z = 30, or in radians $\frac{\pi}{6}$ .

2006 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 5: / Line 5: @@
 <math> \mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6} \quad \mathrm{(E) \ } \frac{7\pi}{6}</math>
-== Solution ==
+== Solutions ==
+== Solution 1 ==
 *For <math>\cos x = 0</math>, x must be in the form of <math>\frac{\pi}{2} + \pi n</math>, where <math>n</math> denotes any [[integer]].
 *For <math>\cos (x+z) = 1 / 2</math>, <math>x + z = \frac{\pi}{3} +2\pi n, \frac{5\pi}{3} + 2\pi n</math>.
@@ Line 11: / Line 12: @@
 <!-- explanation needed-->The smallest possible value of <math>z</math> will be that of <math>\frac{5\pi}{3} - \frac{3\pi}{2} = \frac{\pi}{6} \Rightarrow \mathrm{(A)}</math>.
-Alternative Solution: Since <math>cos(x) = 0</math>, we know that x must equal some multiple of 90. Let us assume x = 90. We want <math>cos(x+z) = 1/2</math>, and by using the property that <math>cos(x) = cos(180-x)</math>, we want x = 60 since <math>cos(60) = \frac{1}{2}</math>. This means that we have <math>x + z = 120</math>, and from this we see that z = 30, or in radians <math>\frac{\pi}{6}</math>.
+== Solution 2 ==
+Since <math>cos(x) = 0</math>, we know that x must equal some multiple of 90. Let us assume x = 90. We want <math>cos(x+z) = 1/2</math>, and by using the property that <math>cos(x) = cos(180-x)</math>, we want x = 60 since <math>cos(60) = \frac{1}{2}</math>. This means that we have <math>x + z = 120</math>, and from this we see that z = 30, or in radians <math>\frac{\pi}{6}</math>.
 == See also ==

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