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Find the maximum value of

7 cos (θ) + 24 sin (θ) + 7,

where θ ∈ R.

 Mar 11, 2023
 #1
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-1

By Cauchy Schwarz,

(7 + 24 + 7)(7 cos t + 24 sin t + 7) <=  (7^2 + 24^2 + 7^2),

so 7 cos t + 24 sin t + 7 <= 337/19.

 

The maximum value is 337/19.

 Mar 11, 2023
 #2
avatar+33616 
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As follows:

 

 Mar 11, 2023
 #3
avatar+397 
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Here's a non calculus approach.

Begin by noticing that

\(7^{2}+24^{2}=49+576=625=25^{2},\)

so 7, 24, 25 can be used to as the sides of a right-angled triangle.

Let the angle opposite the 7 be \(\phi,\)

then

\(\sin\phi=7/25 \text{ and } \cos\phi=24/25.\)

 

\(\displaystyle 7\cos\theta+24\sin\theta+7\\ = 25(\frac{7}{25}\cos\theta+\frac{24}{25}\sin\theta)+7\\ =25(\sin\phi\cos\theta+\cos\phi\sin\theta)+7 \\ =25\sin(\theta+\phi)+7.\)

 

Maximum value will be 25 + 7 = 32 occuring when

 \(\sin(\theta+\phi)=1, \\ \theta=\pi/2-\tan^{-1}(7/24),\\ 0\leq\theta\leq\pi/2.\)

 Mar 11, 2023

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