2005 iTest Problems/Problem 19

Problem

Find the amplitude of $y = 4 \sin (x) + 3 \cos (x)$ .

Solution 1

By the Cauchy-Schwarz Inequality, $(a \cos \theta + b \sin \theta)^2 \le (a^2 + b^2)(\cos^2 \theta + \sin^2 \theta) = a^2 + b^2$ , so $a \cos \theta +b \sin \theta \le \sqrt{a^2+b^2}$ . If $a=b=0$ , then the expression is equal to zero for any angle. Otherwise, we can construct a right triangle with an angle measuring $\theta$ with side lengths $a,b$ and $\sqrt{a^2+b^2}$ . Evaluating the trig functions for this angle yields $\cos \theta=\frac{a}{\sqrt{a^2+b^2}}$ and $\sin \theta=\frac{b}{\sqrt{a^2}{b^2}}$ , and multiplying the cosine by $a$ and the since by $b$ and adding yields that $a\cos \theta +b \sin \theta=\sqrt{a^2+b^2}$ . Plugging in 3 and 4, we find that the maximum value is $5$ . Since this is a trig function with midline $y=0$ , its amplitude is equal to its maximum value, which is $\boxed{5}$ .

2005 iTest (Problems, Answer Key)
Preceded by: Problem 15	Followed by: Problem 17
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2005 iTest Problems/Problem 19

Problem

Solution 1

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