192k views
4 votes
Find the exact value by using a half-angle identity. cosine of five pi divided by twelve.

1 Answer

4 votes

Answer:


\therefore cos((150\°)/(2))=\frac{\sqrt{2-√(3) } }{2} \approx 0.26

Explanation:

The given expression is


cos((5 \pi)/(12))

To find the exact value using identities, we can split the angle in a sum that is equivalent, that is, we rewrite the expression.

Let's rewrite the expression


cos((5 \pi)/(12))=cos((2 \pi)/(12) +(3 \pi)/(12))

We can rewrite this way, because the sum of those fractions gives the original one. Then, we simplify


cos((5 \pi)/(12))=cos((2 \pi)/(12) +(3 \pi)/(12))=cos((\pi)/(6) +(\pi)/(4))

Now, here we need to transfor from radians to degrees, because that way we can obtain half-angles


cos((5 \pi)/(12))=cos((2 \pi)/(12) +(3 \pi)/(12))=cos((\pi)/(6) +(\pi)/(4))=cos((180\°)/(3(2)) +(180\°)/(2(2)) )

Then, we divide each fraction in a way that the final expression contains halves


cos((5 \pi)/(12))=cos((2 \pi)/(12) +(3 \pi)/(12))=cos((\pi)/(6) +(\pi)/(4))=cos((180\°)/(3(2)) +(180\°)/(2(2)) )=cos((60\°)/(2) +(90\°)/(2) )


cos((150\°)/(2))

The half-angle identity is


cos((\theta)/(2))=\sqrt{(1+cos\theta)/(2) }

In this case,
\theta=150\°, replaing it in the identity, we have


cos((\theta)/(2))=\sqrt{(1+cos\theta)/(2) }\\cos((150\°)/(2))=\sqrt{(1+cos150\°)/(2) }

But,
cos150\°=-cos30\°=-(√(3) )/(2), replacing this


cos((150\°)/(2))=\sqrt{(1+cos150\°)/(2) }=\sqrt{(1-(√(3))/(2) )/(2) }\\cos((150\°)/(2))=\sqrt{((2-√(3) )/(2) )/(2) } =\sqrt{(2-√(3) )/(4) } \\\\\therefore cos((150\°)/(2))=\frac{\sqrt{2-√(3) } }{2} \approx 0.26

User Letronje
by
8.6k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories