Arctan – Definition, Formula, and Uses

Arctan

Also, know as arctangent the arctan is the inverse function of the tangent function. Moreover, in this topic, we will define the arctan, its formula and function and some examples related to it.

Arctan Formula

In mathematics, every function has its inverse. Furthermore, a simple operation like multiplication and addition has inverses like division and subtraction.

Besides, it refers to the contrary of the tangent function and we use it to figure out the angle measured from the tangent ratio of the right-angled triangle, chosen by the formula:

tan = opposite ÷ adjacent

Besides, we use the term ‘arc’ for the reason that, when measuring an angle in radians the arc length of the part of a circle divided by the angle (with the vertex in the middle of the circle) equals to the angle measure.

Also, the standard unit of measuring an angle is the radian. Moreover, it is approximately 57 degrees and based on the radius of the circle.

Use of Arctan

The arctan is trigonometric functions and these functions are used to define values related to right triangles. In practical we use these functions to resolve the distance or height of the object which is difficult to measure.

Also, this is resolute using the measure of one angle (not right angle) and the ratio of two sides of the triangle. Besides, we determine trigonometric functions by the sides of the triangle that we use in the ratio of these formulas:

Sine = opposite ÷ hypotenuse

Cosine = adjacent ÷ hypotenuse

Tangent = opposite ÷ adjacent

Most noteworthy, we can use the inverse of these functions to resolute the angle measures when we knew the sides of the triangle. Also, you can use it to decide the measure of an angle when we knew the side opposite and the side adjacent to the angle.

Besides, it has practical application in many fields like landscaping, building, architecture, physics, engineering, and amidst other mathematical and scientific areas.

Most noteworthy, the best method to use the arctan is a scientific calculator. Furthermore, we can locate the arctan button just above the tangent on the calculator. Also, we can use a data table to resolve arctan; but this can be a tiresome and bulky method, however, it is effective if you don’t have a scientific calculator.

Now let’s look at some of the examples of it to find the angle measure.

Examples of Arctan

Let’s look at some examples to understand it more clearly.

Example 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 trigonometric function arctan to determine the angle measure when you know the side opposite the side adjacent to the angle measure that you are trying to find.

So, the equation will be similar to this:

Arctan θ = opposite ÷ adjacent

Arctan θ = 15 ÷ 23 = 0.65

θ = 33 degrees or \(33^o \)

Example 2

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

 Solution

Arctan θ = opposite ÷ adjacent

Arctan θ = 3 ÷ 2 = 1.5

θ = \(56^o\)

Solved Question for You

Question. What will be the value of θ? If the height of the triangle is 6 ft and the base is 25 ft.

1. \(13^o\)
2. \(15^o\)
3. \(11^o\)
4. \(14^o\)

Answer- The correct answer is option A.

Solution:

arctanθ = opposite ÷ adjacent

arctanθ = 6 ÷ 25 =0.24

θ = \(13^o\)

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