+0

0
23
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Find the maximum value of

7 cos (θ) + 24 sin (θ) + 7,

where θ ∈ R.

Mar 11, 2023

#1
-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
+33400
+1

As follows:

Mar 11, 2023
#3
+269
+1

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