Question

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

Answer 1

Answer:

Step-by-step explanation:

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

Answer 2

Answer:

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

Step-by-step explanation:

The given expression is

$cos(\frac{5 \pi}{12})$

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

$cos(\frac{5 \pi}{12})=cos(\frac{2 \pi}{12} +\frac{3 \pi}{12})$

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

$cos(\frac{5 \pi}{12})=cos(\frac{2 \pi}{12} +\frac{3 \pi}{12})=cos(\frac{\pi}{6} +\frac{\pi}{4})$

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

$cos(\frac{5 \pi}{12})=cos(\frac{2 \pi}{12} +\frac{3 \pi}{12})=cos(\frac{\pi}{6} +\frac{\pi}{4})=cos(\frac{180\°}{3(2)} +\frac{180\°}{2(2)} )$

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

$cos(\frac{5 \pi}{12})=cos(\frac{2 \pi}{12} +\frac{3 \pi}{12})=cos(\frac{\pi}{6} +\frac{\pi}{4})=cos(\frac{180\°}{3(2)} +\frac{180\°}{2(2)} )=cos(\frac{60\°}{2} +\frac{90\°}{2} )$

$cos(\frac{150\°}{2})$

The half-angle identity is

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

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

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

But, $cos150\°=-cos30\°=-\frac{\sqrt{3} }{2}$, replacing this

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