Question

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

Answer 1

`\cos(x+(\pi/3))={1+3\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:

`\cos(color(blue)(x)+color(gold)(\pi/3))=color(blue)(\cos(x))color(gold)(\cos(\pi/3))-color(blue)(\sin(x))color(gold)(\(sin\pi/3))`

`=(1/4)(1/2)-({-\sqrt{15}}/4)(\{\sqrt{3}}/2)`

`={1+3\sqrt{5}}/8`