Question

Find the exact value by using a half-angle identity. cosine of five pi divided by twelve.

Answer 1

`\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`