We have #sec(315) = 1/cos(315)#
Assuming #315# is a value in degrees, we can rewrite that to #315º = 360º-45º#, and since #360º# is a full loop, we can say that
#sec(315º) = 1/cos(-45º)#
Since the cosine is an even function, we have that #cos(-x) = cos(x)#, so
#sec(315º) = 1/cos(45º)#
#45º# is a special angle so we know that its cosine is #sqrt(2)/2#, so
#sec(315º) = 1/(sqrt(2)/2) = 2/sqrt(2) = sqrt(2)#
If it's a value in radians, we see that #315 - 100*pi ~= 0.84#, whose cosine is coincidentally #2/3# - or close to it - so
#sec(315) = 1/cos(315) ~= 1/cos(0.84) ~= 3/2#
Which is only a little bit bigger than the proper answer, approximately #1.4994#. But yeah, you can't really solve this without calculator. You could at best give a range based on the special angles, like #sec(pi/6) < sec(315) < sec(pi/3)#
