Respuesta :
Answer:
[tex]\frac{1}{4}[/tex] ([tex]\sqrt{2}[/tex] - [tex]\sqrt{6}[/tex] )
Step-by-step explanation:
Using the addition formula for cosine
cos(a + b) = cosacosb - sinasinb
and the exact values
cos[tex]\frac{\pi }{3}[/tex] = [tex]\frac{1}{2}[/tex], sin[tex]\frac{\pi }{3}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] , cos[tex]\frac{\pi }{4}[/tex] = sin[tex]\frac{\pi }{4}[/tex] = [tex]\frac{\sqrt{2} }{2}[/tex]
Note [tex]\frac{7\pi }{12}[/tex] = [tex]\frac{\pi }{3}[/tex] + [tex]\frac{\pi }{4}[/tex] , thus
cos[tex]\frac{7\pi }{12}[/tex] = cos([tex]\frac{\pi }{3}[/tex] + [tex]\frac{\pi }{4}[/tex] )
= cos[tex]\frac{\pi }{3}[/tex]cos[tex]\frac{\pi }{4}[/tex] - sin[tex]\frac{\pi }{3}[/tex]sin[tex]\frac{\pi }{4}[/tex]
= ( [tex]\frac{1}{2}[/tex] × [tex]\frac{\sqrt{2} }{2}[/tex] ) - ([tex]\frac{\sqrt{3} }{2}[/tex] × [tex]\frac{\sqrt{2} }{2}[/tex] )
= [tex]\frac{\sqrt{2} }{4}[/tex] - [tex]\frac{\sqrt{6} }{4}[/tex]
= [tex]\frac{1}{4}[/tex] ([tex]\sqrt{2}[/tex] - [tex]\sqrt{6}[/tex] )
