How do you evaluate #sec (4pi)#?

1 Answer
Aviv S.
Mar 31, 2018

#sec(4pi)=1#

Explanation:

Use the reciprocal identity:

#sectheta=1/costheta#

Also, since all the trigonometric functions are periodic, you can subtract or add any multiples of #2pi# until the angle is a bit easier to calculate:

#color(white)=sec(4pi)#

#=1/cos(4pi)#

#=1/cos(4picolor(red)-color(red)(2pi))#

#=1/cos(2pi)#

#=1/cos(2picolor(red)-color(red)(2pi))#

#=1/cos(0)#

Here's a unit circle to remind us of some trig values for cosine:

enter image source here

Now we can see that #cos(0)# is #1#, so:

#color(white)=1/cos(0)#

#=1/1#

#=1#

That's the result. We can check our work using a calculator:

https://www.desmos.com/calculator

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