How do you calculate sec((13pi)/4)?

1 Answer
May 11, 2015

Recall the definition of a function sec(phi):
sec(phi)=1/cos(phi)

Recall the definition a function cos(phi):
cos(phi) is an abscissa (X-coordinate) of an endpoint of a radius-vector in the unit circle that forms an angle phi with a positive direction of the X-axis, counting counter-clockwise from the positive direction of the X-coordinate towards a radius-vector.

From this definition of a function cos(phi) follows that
(a) Function cos(phi) is periodical with a period of 2pi.
(b) Function cos(phi) is even in terms of cos(phi)=cos(-phi).

Using these properties, we can state that
cos((13pi)/4)=cos((13pi)/4-4pi)=cos(-(3pi)/4)=cos((3pi)/4)

The angle (3pi)/4 lies in the second quarter and the abscissa of an endpoint of a unit vector that corresponds to this angle is negative.
The corresponding angle with positive but equal by absolute value abscissa is, obviously, pi-(3pi)/4=pi/4.

So, we can conclude that cos((13pi)/4)=-cos(pi/4)=-sqrt(2)/2.

Now we can calculate the value of sec((13pi)/4):
sec((13pi)/4) = 1/cos((13pi)/4) = 1/[-sqrt(2)/2]=-2/sqrt(2)=-sqrt(2)

So, the answer is
sec((13pi)/4) = -sqrt(2)