# If cos(x) = (1/4), where x lies in quadrant 4, how do you find cos(x + pi/3)?

Mar 12, 2017

$\setminus \cos \left(x + \left(\setminus \frac{\pi}{3}\right)\right) = \frac{1 + 3 \setminus \sqrt{5}}{8}$, approximately $0.964$.

#### Explanation:

First off, since $x$ is in quadrant $4$ below the horizontal axis, its sine is negative:

\sin(x)=-\sqrt{1-(1/4)^2}=-\{sqrt{15}}/4}

Then, apply the formula for the cosine of a sum:

$\setminus \cos \left(\textcolor{b l u e}{x} + \textcolor{g o l d}{\setminus \frac{\pi}{3}}\right) = \textcolor{b l u e}{\setminus \cos \left(x\right)} \textcolor{g o l d}{\setminus \cos \left(\setminus \frac{\pi}{3}\right)} - \textcolor{b l u e}{\setminus \sin \left(x\right)} \textcolor{g o l d}{\setminus \left(\sin \setminus \frac{\pi}{3}\right)}$

$= \left(\frac{1}{4}\right) \left(\frac{1}{2}\right) - \left(\frac{- \setminus \sqrt{15}}{4}\right) \left(\setminus \frac{\setminus \sqrt{3}}{2}\right)$

$= \frac{1 + 3 \setminus \sqrt{5}}{8}$