Excel TAN Function: How to Find the Tangent Angle

ATAN, ASIN, or ACOS formulas are your friends in this case

Image of a compass for measuring angles

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If you're working with any triangle that has a right angle in it somewhere, it's very easy to find the tangent angle, so long as you know the length of two sides of the triangle.

This is even easier in Microsoft Excel, because there are built-in functions you can use.

What Is the Tangent Angle?

A tangent angle is the angle in the triangle where you know the length of the side opposite the angle, and the length of the side adjacent to it.

Imagine, for example, that your boss tells you to adjust a ladder at exactly 70 degrees from the ground. Unless you have some special tools, it would be very difficult for you to measure whether the angle between the ladder and the ground is exactly 70 degrees.

A ladder placed up against a wall.

However, if you have a measuring tape, you could measure the distance from the bottom of the ladder to the wall. Since the ladder against the wall forms a triangle, this would be the side that's "adjacent" to the tangent angle you're trying to calculate.

Next you'd measure the distance from the bottom of the wall to where the top of the ladder touches it. This is the distance of the side that's "opposite" from the tangent angle.

An example of a right triangle.

With just the opposite and adjacent sides, you can calculate the angle at the base of the ladder using the arctangent function.

If the wall (opposite) side is 10 feet, and the ground (adjacent) side is 5 feet, the formula for the tangent angle is the opposite side divided by the adjacent side. This is 10 divided by 5, or 0.5.

To find the value for the angle itself, you need to take the arctangent of 0.5.

Finding the Tangent Angle with Excel

You could try to find a calculator that can calculate the arctangent of a value, but Excel has a built-in function called ATAN that you can use.

Example of using Arctangent function in Excel

The formula returns the angle in radians, which your boss probably won't understand.

You'll want to convert radians to degrees by multiplying it by 180/pi.

Luckily, Excel also has a PI function you can use for this purpose.

An example of converting radians to degrees in Excel.

The answer in this case is 63.43 degrees. This means you'll need to adjust one of the lengths until the angle is exactly 70 degrees.

This is easy in Excel, because you can just change the value of the opposite side until the arctangent result is 70.

You've figured out how to place the ladder, and your boss is happy!

Using ASIN and ACOS in Excel

In this same scenario, let's say you don't have a tape measure long enough to measure the wall. You only know that the ladder is 15 feet, and you've measured that it's placed 5 feet from the wall.

How do you calculate the angle?

Thankfully, there are two other functions available in Excel that you can use in this situation.

The length of the ladder is the "hypotenuse" of the triangle, and the ground distance is the adjacent side to the angle. So long as the triangle has one right (90 degree) angle, the information you have determines the formula you need to use.

  • Cosine: You calculate the cosine angle if you know the length of the hypotenuse and the adjacent side.
  • Sine: Calculate the sine angle if you know the length of the hypotenuse and the opposite side.

In this case, the angle is the arccosine of the adjacent side divided by the hypotenuse.

Since you know the adjacent side (the ground distance) is 5 feet, and the ladder length (hypotenuse) is 15 feet, the cosine of the angle is 5 divided by 15, or 0.333.

To calculate the angle, you need to use the arccosine formula in Excel.

The result of the arccosine function is Excel is in radians, so you need to multiply it by 180/PI to convert it to degrees.

In this case, with a 15 foot ladder with its base 5 feet from the wall, the angle is 70.53 degrees.

If you knew that the height of the wall (the opposite side) is 10 feet, instead of the ground distance from the wall (the adjacent side), you'd need to use the arcsine formula in Excel.

In this case, the sine of the angle is the opposite side divided by the hypotenuse.

An example of using ACOS function in Excel.

After converting to degrees, the angle in this case would be 48.12 degrees.

Why Use ATAN, ACOS, or ASIN?

You may be surprised just how often situations come up where you may need to use one of these functions in Excel.

Just a few examples include:

  • In carpentry and construction, angles and lengths are used in all aspects of building houses and buildings.
  • Photographers use angles to carefully align lighting and their creative shots.
  • In sports, understanding angles can enhance skill and improve strategy.
  • Ships and airplanes are located on radar using angles and distances.
  • If you want to be sure furniture is going to fit right in your room, you're going to need to know how to calculate lengths and angles!

You may be able to accomplish these calculations on a scientific calculator. But if you don't have one handy, Excel can help you make those calculations better than anything else.