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## 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:

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.

## 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 |

f(x) |

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:

**SEC(number)**

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.

## References

Office Support: SEC Function. Retrieved May 28, 2019 from: https://support.office.com/en-ie/article/sec-function-ff224717-9c87-4170-9b58-d069ced6d5f7

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**Stephanie Glen**. "Secant Function (Sec Function)" From

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