Calculus How To

Secant Function (Sec Function)

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Contents (Click to skip to that section):

  1. What is the Secant Function?
  2. Limit of a Secant Function
  3. SEC function in Excel

What is a Secant Function?

The secant function (also called the sec function) is the reciprocal function of the cosine function. It is defined as:
secant function formula

The domain for the secant function is all values for which cos≠0. In other words, for all odd half integer multiples of π. At these points, the function has vertical asymptotes. The period for this function is 2π.

graph of secant function

Graph of the secant function.

Limit of a Secant Function

Finding the limit of a secant function can seem imposing when you look at a graph of the function, but approaching the limit in small steps (by making a table) makes it relatively simple. One key fact to keep in mind is that if a limit does not approach the same value from the left and the right, then the limit does not exist.

Example Problem: Find the limit of sec (x) as x approaches 1.

Step 1: Draw a table . Include a few small values around your x-value.

x 0.70 0.90 0.99 c=1.00 1.01 1.10 1.30

Step 2: Evaluate for the value of the function at the x values you set up, using a calculator. I used Google for these calculations (for example, type in “secant(0.70 degrees)”). If you get don’t get the answers I did below, make sure you’re using degrees, and not radians.

x 0.70 0.90 0.99 c=1.00 1.01 1.10 1.30
f(x) 1.00007 1.00012 1.00015 1.00015 1.00016 1.00018 1.0026

Step 3: Analyze the calculated values to find the limit. If the limit isn’t clear, but the values do seem to be heading towards a limit from both sides, continue to use numbers closer and closer to c to narrow down the limit.
In this example, the values are trending towards 1.00015.

Example Problem: Find the limit of sec (x) as x approaches 90.

Step 1: Repeat the above steps for the new values.

x 89.70 89.70 89.99 c=0.90 90.01 90.10 90.30
f(x) 190.99 572.96 5729.5 Undefined -5729.5 -572.96 -190.99

The function is undefined at 90, and approaching 90 from the left tends towards infinity, while approaching 90 from the right tends towards negative infinity. In this case, the limit of a secant does not exist. For the secant function, this will occur at 90 and at every interval of 180 either direction from it.

SEC Function in Excel

The SEC function in Excel gives you the secant of a certain angle. The format is:
Where “number” is the angle (in radians) for which you want the secant.
Note: If your angle is in degrees, Google will convert it for you.

For example, =SEC(45) gives you the secant of a 45 degree angle, which is 1.90359.


Office Support: SEC Function. Retrieved May 28, 2019 from:

Stephanie Glen. "Secant Function (Sec Function)" From Calculus for the rest of us!

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