How do I find the value of sec 225?

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Answer 1 · 2015-09-17T14:12:43

#sec225 = -sqrt2#

First thing we do is remember that #secx = 1/cosx#, so

#sec225 = 1/cos225#

Then we see that we can rewrite 225 as #180 + 45#, so

#sec225 = 1/cos(180+45)#

Using the formula #cos(a+b) = cos(a)cos(b) - sin(a)sin(b)# we have that

#cos225 = cos(180)cos(45) - sin(180)sin(45)#

From looking at the unit circle we know that #cos(180) = -1# and that #sin(180) = 0#, so

#cos225 = (-1)cos(45) - 0sin(45)#
#cos225 = -cos45#

We know that #cos(45) = sqrt2/2#, so

#cos225 = -sqrt2/2#
Therefore the secant is

#sec225 = 1/cos225 = -2/sqrt2#

Rationalizing,

#sec225 = -2sqrt2/2#

And finally we cancel that pesky 2

#sec225 = -sqrt2#