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Difference between revisions of "2002 AMC 12P Problems/Problem 17"

(→‎Solution 2 (Cheese))
(→‎Solution 1)
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==Solution 1==
 
==Solution 1==
 +
We can simplify the expression using the Pythagorean identity, <math>\sin^2 {x} + \cos^{x}=1.</math>
 +
 
== Solution 2 (Cheese)==
 
== Solution 2 (Cheese)==
 
We don't actually have to solve the question. Just let <math>x</math> equal some easy value to calculate <math>\cos {x}, \cos {2x}, \sin {x}, \sin {\frac{x}{2}},</math> and <math>\cos {\frac{x}{2}}.</math> For this solution, let <math>x=60^\circ.</math> This means that the expression in the problem will give <math>\sqrt{\sin^4{60^\circ} + 4 \cos^2{60^\circ}} - \sqrt{\cos^4{60^\circ} + 4 \sin^2{60^\circ}}=\sqrt{(\frac{\sqrt{3}}{2})^4 + 4(\frac{1}{2})^2} - \sqrt{(\frac{1}{2})^4 + 4(\frac{\sqrt{3}}{2})^2}=\sqrt{\frac{9}{16} +1} - \sqrt{\frac{1}{16} + 3} = \frac{-1}{2}.</math> Plugging in <math>x=60^\circ</math> for the rest of the expressions, we get
 
We don't actually have to solve the question. Just let <math>x</math> equal some easy value to calculate <math>\cos {x}, \cos {2x}, \sin {x}, \sin {\frac{x}{2}},</math> and <math>\cos {\frac{x}{2}}.</math> For this solution, let <math>x=60^\circ.</math> This means that the expression in the problem will give <math>\sqrt{\sin^4{60^\circ} + 4 \cos^2{60^\circ}} - \sqrt{\cos^4{60^\circ} + 4 \sin^2{60^\circ}}=\sqrt{(\frac{\sqrt{3}}{2})^4 + 4(\frac{1}{2})^2} - \sqrt{(\frac{1}{2})^4 + 4(\frac{\sqrt{3}}{2})^2}=\sqrt{\frac{9}{16} +1} - \sqrt{\frac{1}{16} + 3} = \frac{-1}{2}.</math> Plugging in <math>x=60^\circ</math> for the rest of the expressions, we get

Revision as of 01:29, 31 December 2023

Problem

Let $f(x) = \sqrt{\sin^4{x} + 4 \cos^2{x}} - \sqrt{\cos^4{x} + 4 \sin^2{x}}.$ An equivalent form of $f(x)$ is

$\text{(A) }1-\sqrt{2}\sin{x} \qquad \text{(B) }-1+\sqrt{2}\cos{x} \qquad \text{(C) }\cos{\frac{x}{2}} - \sin{\frac{x}{2}} \qquad \text{(D) }\cos{x} - \sin{x} \qquad \text{(E) }\cos{2x}$

Solution

Solution 1

We can simplify the expression using the Pythagorean identity, $\sin^2 {x} + \cos^{x}=1.$

Solution 2 (Cheese)

We don't actually have to solve the question. Just let $x$ equal some easy value to calculate $\cos {x}, \cos {2x}, \sin {x}, \sin {\frac{x}{2}},$ and $\cos {\frac{x}{2}}.$ For this solution, let $x=60^\circ.$ This means that the expression in the problem will give $\sqrt{\sin^4{60^\circ} + 4 \cos^2{60^\circ}} - \sqrt{\cos^4{60^\circ} + 4 \sin^2{60^\circ}}=\sqrt{(\frac{\sqrt{3}}{2})^4 + 4(\frac{1}{2})^2} - \sqrt{(\frac{1}{2})^4 + 4(\frac{\sqrt{3}}{2})^2}=\sqrt{\frac{9}{16} +1} - \sqrt{\frac{1}{16} + 3} = \frac{-1}{2}.$ Plugging in $x=60^\circ$ for the rest of the expressions, we get

$\text{(A) }1-\sqrt{2}\sin{60^\circ}=1-\frac{\sqrt{6}}{2} \neq \frac{-1}{2}.$

$\text{(B) }1+\sqrt{2}\cos{60^\circ}=1+\frac{\sqrt{2}}{2} \neq \frac{-1}{2}.$

$\text{(C) }\cos{\frac{60^\circ}{2}} - \sin{\frac{60^\circ}{2}}=\frac{\sqrt{3}}{2}-1 \neq \frac{-1}{2}.$

$\text{(D) }\cos{60^\circ} - \sin{60^\circ}=\frac{1-\sqrt{3}}{2} \neq \frac{-1}{2}.$

$\text{(E) }\cos{2(60^\circ)}=\frac{-1}{2}.$

Therefore, our answer is $\boxed{\textbf{(E) } \cos {2x}}$.

See also

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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

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