Question

IF COS X = 1/4and π/2< x < 1, find the exact value ofcos( π/6 - x)

Answer 1

From the question;

we are given

$\cos \text{ x = }\frac{1}{4},\text{ }\frac{\pi}{2}<\text{ x }<1$

we are to find

$cos\text{ (}\frac{\pi}{6}\text{ - x)}$

To find this we will need to simplify

therefore

$\begin{gathered} \sin ce\text{ }\frac{\pi}{6}=\text{ }\frac{180}{6}\text{ =30} \\ \text{hence} \\ \cos (\frac{\pi}{6}\text{ - x) = cos (30 - x)} \end{gathered}$

applying trigonometric formula

$\cos (A\text{ - B) = cosAcosB + sinAsinB}$

Hence

$\cos (30\text{ - x) = cos30cosx + sin30sinx}$

First, we need to find sinx

Starting with cos x = 1/4

we will apply trigonemetric ratio

SOH CAH TOA

Therefore

$\cos \text{ x = }\frac{\text{adjacent}}{\text{hypothenus}}$

This gives

adjacent = 1

hypothenus = 4

constructing the triangle we get

From triangle

Applying pythagoras rule

$\begin{gathered} \text{hyp}^2=opp^2+adj^2 \\ \text{therefore} \\ 4^2=opp^2+1^2 \\ 16=opp^2\text{ + 1} \\ \text{opp}^2\text{ = 16 - 1} \\ \text{opp}^2\text{ = 15} \\ \text{opp =}\sqrt[]{15} \end{gathered}$

finding sin x

using trigonometric ratio

$\begin{gathered} \sin \text{ x = }\frac{\text{opposite}}{hypotenus} \\ \text{opp =}\sqrt[]{15},\text{ hypotenus = 4} \\ \text{therefore} \\ \sin \text{ x = }\frac{\sqrt[]{15}}{4} \end{gathered}$

Finally

Finding

$\cos (\frac{\pi}{6}-x)$

recall

$\begin{gathered} \cos (\frac{\pi}{6}-x)\text{ = cos(30 - x)} \\ \text{and} \\ \cos (30-x)\text{ = cos30cosx + sin30sinx} \end{gathered}$

Therefore

appying our values we have

$\begin{gathered} \cos (30\text{ - x) = cos30cosx + sin30sinx} \\ \cos (30\text{ - x) = }\frac{\sqrt[]{3}}{2}\times\frac{1}{4}\text{ + }\frac{1}{2}\times\frac{\sqrt[]{15}}{4} \\ \cos (30\text{ - x) = }\frac{\sqrt[]{3}}{8}\text{ + }\frac{\sqrt[]{15}}{8} \\ \cos (30\text{ - x) = }\frac{\sqrt[]{3}\text{ + }\sqrt[]{15}}{8} \end{gathered}$

Therefore

The answer is

$\cos (\frac{\pi}{6}-x)\text{ = }\frac{\sqrt[]{3}+\text{ }\sqrt[]{15}}{8}$