# Arctan: Definition, Formula, Applications, Solved Examples & More

*Arctan Formula*

**Definition:** Arctan is the inverse function of the tangent function which is the ratio of the side opposite an angle divided by the side adjacent to the angle (i.e., tan = opposite/adjacent). It is also known as arctangent.

**Tan Inverse or Arctan Formula**

The basic arctan formula is given as follows:

Tan (θ)= Opposite Side/Adjacent Side

θ = Tan^{-1} (Opposite Side/Adjacent Side)

Or θ = arctan (Opposite Side/Adjacent Side)

where θ is an angle

In the inverse trigonometric function arctan, the term arc is used because when measuring an angle in radians, the arc length of a part of the circle bisected by an angle (with the vertex at the center of the circle) equals the angle measure.

The radian is the standard unit of measurement for an angle and is equal to approximately 57 degrees. It is based on the radius of a circle.

**Use of Arctan formula**

Arctan is generally used to compute the angle measure from the tangent ratio of a right triangle. Arctan can be calculated in terms of degrees and as well as radians.

The arctan being a trigonometric function is used to define values related to right triangles. In practical, these functions can be used to determine heights of objects or distances that are difficult to measure. These measurements can be determined using the measure of one angle other than the right angle and a ratio of two sides of the triangle.

**1. ****In the right triangle, the base is 23 m and the height is 15 m. You have to find the angle theta (θ) that is the opposite of the right angle.**

**Solution:**

We can use the inverse trigonometric function arctan to determine the angle measure.

Using arctan formulaengi

θ = arctan(opposite ÷ adjacent)

⇒ θ = arctan(15 ÷ 23)

⇒ θ = arctan (0.65)

⇒ θ = 33 degrees

**2. In the right-angle triangle XYZ, the base is 2 inch and the height is 3 inches. find the value of θ that is the opposite of the right angle.**

**Solution:**

Using arctan formula:

θ = arctan(opposite ÷ adjacent)

θ = arctan(3 ÷ 2)

θ = arctan1.5

θ = 56 degrees

So, the required angle is 56^{o}.

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