# Tan-1 Calculator

Created by Anna Szczepanek, PhD
Reviewed by Davide Borchia
Last updated: Jan 20, 2023

The tan-1 calculator is here to help whenever you struggle with equations of the form tan-1(x) = y. We're here to help you understand what these equations mean and how to solve them.

## What is tan-1 in math?

The notation tan-1 may mean two different things in math:

• tan-1(x) = 1/tan(x) = cot(x), i.e., we deal here with the multiplicative inverse; or
• tan-1(x) = arctan(x), so the inverse function of the tangent. We're answering here the question of what is the angle whose tangent is equal to x.

People who write tan-1 most often have in mind the latter meaning: why would you bother typing tan-1(x) if you can just write cot(x)? However, sometimes you'll need to guess from the context.

As you can see, the notation tan-1 can be extremely confusing and you should avoid it. If you mean the cotangent, use cot(x). What should you use for the inverse of tangent? We'll discuss it now.

## What is the notation for the inverse of tangent?

The most common notation for the inverse of the tangent is arctan(x). The prefix arc has its roots in the fact that when using a unit circle and measuring angles in radians, an angle of x radians will correspond to an arc whose length is also x. Hence, "the angle whose tangent is x" coincides with "the arc whose tangent is x".

In programming languages, the inverse of tangent is often shortened to atan(x).

## How to use this tan-1 calculator?

This tan-1 calculator is a very straightforward tool to use. Just input a number in the field and enjoy seeing the results in the blink of an eye!

For instance, if you need to determine tan-1(2), just type in 2 in the field x. You'll see that the result is 1.2490 radians, so a bit more than 71.5°. Note that the calculator allows you to *convert between radians and degrees, you don't need to do that by hand or look for additional tools!

Besides this tan-1 calculator, Omni features several other tools explaining the inverse of tangent from different angles (pun intended). Here they are:

• Arctan calculator;
• Tan inverse calculator;
• Arcus tangent calculator; and
• Inverse tangent calculator.

## FAQ

### How do I find the tan-1 of negative numbers?

To determine the tan-1 of a negative number, follow these steps:

1. Write down the absolute value of your number. In other words, get rid of the minus sign.
2. Find the tan-1 of the absolute value you've found in Step 1.
3. Take the result and write the minus sign in front of it. Formally: find the opposite number.
4. This is your result! We've applied here the formula tan-1(-x) = -tan-1(x) for every real number x.

### What is the tan-1 of -1?

The answer is -45°, or equivalently, -π/4 rad. To arrive at this result, you must use the formula tan-1(-x) = -tan-1(x). Plugging in x = -1, we obtain tan-1(-1) = - tan-1(1) and so it suffices to recall that tan(π/4) = 1.

Anna Szczepanek, PhD
y = arctan(x)
x
y
deg
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