How do you evaluate sec (4pi)?

1 Answer
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/calculatorhttps://www.desmos.com/calculator