Respuesta :
Answer:
[tex]\therefore cos(\frac{150\°}{2})=\frac{\sqrt{2-\sqrt{3} } }{2} \approx 0.26[/tex]
Step-by-step explanation:
The given expression is
[tex]cos(\frac{5 \pi}{12})[/tex]
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
[tex]cos(\frac{5 \pi}{12})=cos(\frac{2 \pi}{12} +\frac{3 \pi}{12})[/tex]
We can rewrite this way, because the sum of those fractions gives the original one. Then, we simplify
[tex]cos(\frac{5 \pi}{12})=cos(\frac{2 \pi}{12} +\frac{3 \pi}{12})=cos(\frac{\pi}{6} +\frac{\pi}{4})[/tex]
Now, here we need to transfor from radians to degrees, because that way we can obtain half-angles
[tex]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)} )[/tex]
Then, we divide each fraction in a way that the final expression contains halves
[tex]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} )[/tex]
[tex]cos(\frac{150\°}{2})[/tex]
The half-angle identity is
[tex]cos(\frac{\theta}{2})=\sqrt{\frac{1+cos\theta}{2} }[/tex]
In this case, [tex]\theta=150\°[/tex], replaing it in the identity, we have
[tex]cos(\frac{\theta}{2})=\sqrt{\frac{1+cos\theta}{2} }\\cos(\frac{150\°}{2})=\sqrt{\frac{1+cos150\°}{2} }[/tex]
But, [tex]cos150\°=-cos30\°=-\frac{\sqrt{3} }{2}[/tex], replacing this
[tex]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[/tex]
