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Difference between revisions of "1984 AHSME Problems/Problem 15"

(Created page with "==Problem 15== If <math> \sin{2x}\sin{3x}=\cos{2x}\cos{3x} </math>, then one value for <math> x </math> is <math> \mathrm{(A) \ }18^\circ \qquad \mathrm{(B) \ }30^\circ \qquad \...")
 
(Solution 3)
 
(4 intermediate revisions by 3 users not shown)
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We divide both sides of the [[equation]] by <math> \cos{2x}\times\cos{3x} </math> to get <math> \frac{\sin{2x}\times\sin{3x}}{\cos{2x}\times\cos{3x}}=1 </math>, or <math> \tan{2x}\times\tan{3x}=1 </math>.
 
We divide both sides of the [[equation]] by <math> \cos{2x}\times\cos{3x} </math> to get <math> \frac{\sin{2x}\times\sin{3x}}{\cos{2x}\times\cos{3x}}=1 </math>, or <math> \tan{2x}\times\tan{3x}=1 </math>.
  
−
This looks a lot like the formula relating the slopes of two [[perpendicular]] [[Line|lines]], which is <math> m_1\timesm_2=-1 </math>, where <math> m_1 </math> and <math> m_2 </math> are the [[Slope|slopes]]. It's made even more relatable by the fact that the [[tangent]] of an [[angle]] can be defined by the slope of the line that makes that angle with the [[x-axis]].
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This looks a lot like the formula relating the slopes of two [[perpendicular]] [[Line|lines]], which is <math> m_1\times m_2=-1 </math>, where <math> m_1 </math> and <math> m_2 </math> are the [[Slope|slopes]]. It's made even more relatable by the fact that the [[tangent]] of an [[angle]] can be defined by the slope of the line that makes that angle with the [[x-axis]].
  
 
We can make this look even more like the slope formula by multiplying both sides by <math> -1 </math>:
 
We can make this look even more like the slope formula by multiplying both sides by <math> -1 </math>:
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label("$3x$",(.125,-.06),ESE);
 
label("$3x$",(.125,-.06),ESE);
 
</asy>
 
</asy>
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Since the product of the two slopes, <math> \tan{2x} </math> and <math> \tan{-3x} </math>, is <math> -1 </math>, the lines are perpendicular, and the [[angle]] between them is <math> \frac{\pi}{2} </math>. The angle between them is also <math> 2x+3x=5x </math>, so <math> 5x=\frac{\pi}{2} </math> and <math> x=\frac{\pi}{10} </math>, or <math> 18^\circ, \boxed{\text{A}} </math>.
 
Since the product of the two slopes, <math> \tan{2x} </math> and <math> \tan{-3x} </math>, is <math> -1 </math>, the lines are perpendicular, and the [[angle]] between them is <math> \frac{\pi}{2} </math>. The angle between them is also <math> 2x+3x=5x </math>, so <math> 5x=\frac{\pi}{2} </math> and <math> x=\frac{\pi}{10} </math>, or <math> 18^\circ, \boxed{\text{A}} </math>.
  
 
NOTE: To show that <math> 18^\circ </math> is not the only solution, we can also set <math> 5x </math> equal to another angle measure congruent to <math> \frac{\pi}{2} </math>, such as <math> \frac{5\pi}{2} </math>, yielding another solution as <math> \frac{\pi}{2}=90^\circ </math>, which clearly is a solution to the equation.
 
NOTE: To show that <math> 18^\circ </math> is not the only solution, we can also set <math> 5x </math> equal to another angle measure congruent to <math> \frac{\pi}{2} </math>, such as <math> \frac{5\pi}{2} </math>, yielding another solution as <math> \frac{\pi}{2}=90^\circ </math>, which clearly is a solution to the equation.
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==Solution 2==
 +
We can simply try the answers. We quickly see that <math> 18^\circ</math> works, since <math>\sin(36^\circ)\sin(54^\circ)=\cos(54^\circ)\cos(36^\circ)=\cos(36^\circ)\cos(54^\circ)</math> by the identity <math>\cos(x)=\sin(90^\circ-x)</math>
 +
==Solution 3==
 +
Start by subtracting <math>\sin{2x}\sin{3x}</math> from both sides to get <math>\cos{2x}\cos{3x}-\sin{2x}\sin{3x}=0</math>. We recognize that this is of the form <math>\cos{a}\cos{b}-\sin{a}\sin{b}=\cos{(a+b)}</math>, so <math>\cos{2x}\cos{3x}-\sin{2x}\sin{3x}=\cos{(2x+3x)}=0</math>. <math>\cos{90^\circ}=0</math>, so <math>x=\boxed{18^\circ}</math>.
  
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~purplepenguin2
 
==See Also==
 
==See Also==
 
{{AHSME box|year=1984|num-b=14|num-a=16}}
 
{{AHSME box|year=1984|num-b=14|num-a=16}}
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{{MAA Notice}}

Latest revision as of 18:16, 4 June 2021

Problem 15

If $\sin{2x}\sin{3x}=\cos{2x}\cos{3x}$, then one value for $x$ is

$\mathrm{(A) \ }18^\circ \qquad \mathrm{(B) \ }30^\circ \qquad \mathrm{(C) \ } 36^\circ \qquad \mathrm{(D) \ }45^\circ \qquad \mathrm{(E) \ } 60^\circ$

Solution

We divide both sides of the equation by $\cos{2x}\times\cos{3x}$ to get $\frac{\sin{2x}\times\sin{3x}}{\cos{2x}\times\cos{3x}}=1$, or $\tan{2x}\times\tan{3x}=1$.

This looks a lot like the formula relating the slopes of two perpendicular lines, which is $m_1\times m_2=-1$, where $m_1$ and $m_2$ are the slopes. It's made even more relatable by the fact that the tangent of an angle can be defined by the slope of the line that makes that angle with the x-axis.

We can make this look even more like the slope formula by multiplying both sides by $-1$:

$\tan{2x}\times-\tan{3x}=-1$, and using the trigonometric identity $-\tan{x}=\tan{-x}$, we have $\tan{2x}\times\tan{-3x}=-1$.

Now it's time for a diagram:

[asy] unitsize(2.54cm); draw(unitcircle); draw((0,-1.25)--(0,1.25)); draw((-1.25,0)--(1.25,0)); draw((0,0)--(cos(2pi/10),sin(2pi/10))); draw((0,0)--(cos(-3pi/10),sin(-3pi/10))); label("$\tan{-3x}$",(cos(-3pi/10),sin(-3pi/10)),SE); label("$\tan{2x}$",(cos(2pi/10),sin(2pi/10)),NE); label("$2x$",(.125,.03),ENE); label("$3x$",(.125,-.06),ESE); [/asy] Since the product of the two slopes, $\tan{2x}$ and $\tan{-3x}$, is $-1$, the lines are perpendicular, and the angle between them is $\frac{\pi}{2}$. The angle between them is also $2x+3x=5x$, so $5x=\frac{\pi}{2}$ and $x=\frac{\pi}{10}$, or $18^\circ, \boxed{\text{A}}$.

NOTE: To show that $18^\circ$ is not the only solution, we can also set $5x$ equal to another angle measure congruent to $\frac{\pi}{2}$, such as $\frac{5\pi}{2}$, yielding another solution as $\frac{\pi}{2}=90^\circ$, which clearly is a solution to the equation.

Solution 2

We can simply try the answers. We quickly see that $18^\circ$ works, since $\sin(36^\circ)\sin(54^\circ)=\cos(54^\circ)\cos(36^\circ)=\cos(36^\circ)\cos(54^\circ)$ by the identity $\cos(x)=\sin(90^\circ-x)$

Solution 3

Start by subtracting $\sin{2x}\sin{3x}$ from both sides to get $\cos{2x}\cos{3x}-\sin{2x}\sin{3x}=0$. We recognize that this is of the form $\cos{a}\cos{b}-\sin{a}\sin{b}=\cos{(a+b)}$, so $\cos{2x}\cos{3x}-\sin{2x}\sin{3x}=\cos{(2x+3x)}=0$. $\cos{90^\circ}=0$, so $x=\boxed{18^\circ}$.

~purplepenguin2

See Also

1984 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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All AHSME Problems and Solutions

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