# Tangent Calculator

This tangent calculator will help you to **find the tangent of any angle** you want. Just enter the angle in degrees or radians, and the tangent will appear in no time. If you want to understand **what a tangent is**, and you're curious about the tangent definition or the tangent formula derivation, keep reading! Also *sin over cos* meme is waiting for you down there.

## What is tangent? Tangent definition, tangent formula

Tangent is one of the three most common trigonometric functions (along with sine and cosine). It may be defined as:

*The ratio of the sides: opposite and adjacent to an angle in a right-angled triangle.*

Check out our ratio calculator to learn more about ratios!

BUT! There's also another geometric concept named *tangent*. It's a line or plane that touches a curve or curved surface at a point, but if extended, it doesn't cross it at that point.

The word *tangent* comes from the Latin *tangere* which means 'to touch'. And our trigonometric function name also comes from that meaning! Have a look at the picture below:

Draw a **unit circle**. Point C is the intersection of the line containing the radius and the line x = 1. Then, tan(α) is simply the y-coordinate of point C.

## Tangent – sin over cos

The tangent of an angle may also be defined as its sine divided by its cosine. But why is it so? Have a look at the picture of a unit circle, and all should be clear:

Knowing the definition of sine – *opposite over hypotenuse* – we can find out that for the right triangle from the image, the `sin(α) = y`

. Analogically, cosine may be defined as *adjacent over hypotenuse*, so in our case, it's equal to x. Then, in the previous section, you learned that the tangent is equal to *opposite over adjacent* side. So, for our example, **tan(α) = y/x**, and that can be substituted by the sine and cosine of our angle of interest to get the final formula:

`tan(α) = sin(α) / cos(α)`

Not such a long time ago, you could find math jokes on the Internet – **sin over cos memes**. We're pretty sure that after reading this paragraph, you'll get this one!

What about **tan**gerine 🍊? **Tan**go 🎶? Ti**tan**ic 🚢? Or even sa**tan**ism? Think about other words which could make such a rebus!

## Law of tangents

The law of tangents describes the relationship between the tangent of two angles of a triangle and the lengths of the opposite sides. Specifically, it states that:

`(a - b) / (a + b) = tan(0.5(α - β)) / tan(0.5(α + β))`

Although the law of tangents is not as popular as the law of sines or the law of cosines, it may be useful when we have given:

- Two sides and the angle between them; and
- Two angles and a side.

🙋 Visit our law of cosines calculator and law of sines calculator if you don't quite remember what these theorems are about!

## Tangent calculator – example of use

All you need to do is **type one value into the calculator – the angle**, in radians or degrees. To change between units, click on the unit name and choose from a drop-down list. Then, you'll see the result immediately – the tangent value of your angle of interest.

Remember that **tan(α) may be undefined**. This situation occurs when cos(α) is 0 because we can never divide by zero (other explanation: the lines will be parallel, so they'll never cross each other, and they won't form the point C).