How can the maximum value of $f(\theta) = 5\cos\theta + 3\cos(\theta + \pi /3)$ be $10$?

Question

My book states that the maximum value of $f(\theta)=5\cos\theta + 3\cos( \theta + \pi /3) $ is $10$.

If I think, no value of $\theta $ exists for which $\cos\theta $ is more than $1$. So how can the maximum value of the above function be $10$? At least I think it should have a maximum value of less than $8$.

heres the solution from my book -

1: https://i.stack.imgur.com/tSmK1.jpg here's the question :-

Answer 1

The book is not saying that the maximum value is $10$, it's only saying the maximum value is no greater than $10$. Similarly, the minimum isn't necessarily $-4$, it's just no less than that.

However, as the OP points out in comments (and I should have seen for myself), the solution is bounding not $5\cos\theta+3\cos(\theta+\pi/3)$, but $5\cos\theta+3\cos(\theta+\pi/3)+3$. The "easy" bounds there are $5+3+3=11$ and $-5-3+3=-5$, so $10$ and $-4$ are definitely better. It appears the book simply neglected to include the additional $+3$ in the problem statement.

Added later: Actually, $10$ and $-4$ are the max and min values. In general, any expression of the form $a\cos\theta+b\sin\theta$ has $\pm\sqrt{a^2+b^2}$ as its exact max and min. I should have noticed this as well.

Answer 2

We have $f( \theta) \le 5+3=8<10$ for all $\theta$.

Answer 3

To get the true extremal values you could also differentiate $f(\theta)$ and find the values of $\theta$ where $f'(\theta)=0$, and then plug these values back into $f(\theta)$.