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

User Kevin Upton
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1 Answer

17 votes
17 votes

From the question;

we are given


\cos \text{ x = }(1)/(4),\text{ }(\pi)/(2)<\text{ x }<1

we are to find


cos\text{ (}(\pi)/(6)\text{ - x)}

To find this we will need to simplify

therefore


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

recall


\begin{gathered} \cos ((\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}*(1)/(4)\text{ + }(1)/(2)*\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 ((\pi)/(6)-x)\text{ = }\frac{\sqrt[]{3}+\text{ }\sqrt[]{15}}{8}

IF COS X = 1/4and π/2< x < 1, find the exact value ofcos( π/6 - x)-example-1
User Antonio Favata
by
2.6k points