Find the Interval of Values of 5 cos x + 3 cos(x + π/3) + 3
Question:
Find the interval in which
\[
5\cos x+3\cos\left(x+\frac{\pi}{3}\right)+3
\]
lies.
Solution
\[ \cos\left(x+\frac{\pi}{3}\right) = \cos x\cos\frac{\pi}{3} – \sin x\sin\frac{\pi}{3} \]
\[ = \frac12\cos x-\frac{\sqrt3}{2}\sin x \]
Therefore, \[ 5\cos x+3\cos\left(x+\frac{\pi}{3}\right)+3 \]
\[ = 5\cos x + \frac32\cos x – \frac{3\sqrt3}{2}\sin x +3 \]
\[ = \frac{13}{2}\cos x – \frac{3\sqrt3}{2}\sin x +3 \]
Maximum value of \[ a\cos x+b\sin x \] is \[ \sqrt{a^2+b^2} \]
\[ = \sqrt{ \left(\frac{13}{2}\right)^2 + \left(\frac{3\sqrt3}{2}\right)^2 } \]
\[ = \sqrt{ \frac{169}{4}+\frac{27}{4} } = \sqrt{\frac{196}{4}} = 7 \]
Hence, \[ -7 \le \frac{13}{2}\cos x – \frac{3\sqrt3}{2}\sin x \le 7 \]
Adding 3 throughout, \[ -4 \le 5\cos x+3\cos\left(x+\frac{\pi}{3}\right)+3 \le 10 \]
\[ \boxed{[-4,\ 10]} \]