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)
