Final answer:
The expansion used for cos(310°) using half-angle identities is cos(310°/2) = -√((1+cos(310°))/2), considering that 310° is in the fourth quadrant.
Step-by-step explanation:
To find the cos(310°) using half-angle identities, we look for an expansion that suits the given angle. Since 310° is in the fourth quadrant, we can use trigonometric identities to find the cosine of half this angle.
The half-angle identity for cosine is: cos(θ/2) = ±√((1+cos(θ))/2), where the sign depends on the quadrant in which θ/2 lies.
As 310°/2 equals 155°, which is in the second quadrant where cosine is negative, we choose the negative square root for our calculation.
Therefore, the expansion for finding cos(310°) would be:
cos(310°/2) = -√((1+cos(310°))/2).