Trigonometry is a branch that deals with the evaluation of sides and angles of a right-angled triangle. The trigonometric operations involve the calculation for sides, angles, and trigonometric ratios. The trigonometric ratios are defined as the values of trigonometric functions which are derived from the ratios of sides and angles of the given triangle. Trigonometry is known to consist of six basic trigonometric functions which are sine, cosine, tangent, cotangent, cosecant, and sectant. 

Inverse Functions

Similarly, as we have the trigonometric functions we have six inverse trigonometric functions written as sin^-1x, cos^-1x, tan^-1x, cosec^-1x, sec^-1x, and cot^-1x. And, the inverse of these trigonometric functions is represented with the prefix ‘arc-‘ such as arcsin, arccos, arctan, arccot, arcsec, and arccosec.

Here, it is better to understand that inverse trigonometric functions are not the reciprocal of their respective trigonometric function. The inverse functions are used to determine the value of an unknown angle by using corresponding trigonometric ratios of it.

What is the Arctan Function in Trigonometry?

In trigonometry, arctan is defined as the inverse function of its respective trigonometric function tangent or inverse tangent function. The inverse tangent function is written with the prefix ‘-arc’ and mathematically, it is represented by tan^-1x.

The arctan gives us the value of the angle by the ratio of perpendicular by the base (perpendicular/base).

Suppose, the tangent of the angle θ equals x.

Then, x = tanθ  

=> θ = tan^-1x

In the given right angle triangle QPR,

=>tanθ = perpendicular/base

=>θ = tan^-1(perpendicular/base)

Table for Arctan

As the value of functions or inverse functions can be evaluated in degrees as well as radians. The value for arctan is given below in the table with respect to the input.

Sample Questions

Question 1. Define arc function in trigonometry.

Answer:

Arc Functions are the inverse trigonometric functions that give the length of the arc for a given value of trigonometric functions.

Question 2. What is arcsin function in trigonometry?

Answer:

The inverse of trigonometric function sine is known as the arcsin function. Mathematically, it is represented as sin^-1x.

Question 3. Mention any four arctan formulas for π.

Answer:

The arctan formulas for π are

• π/4 = 4 arctan(1/5) – arctan(1/239)
• π/4 =  arctan(1/2) + arctan(1/3)
• π/4 = 2 arctan(1/2) – arctan(1/7)
• π/4 = 2 arctan(1/3) +arctan(1/7)

Question 4. List down the standard arctan formulas?

Answer:

The standard arctan formulas are given below:

• θ =arctan(perpendicular/base)
• arctan(-x)=-arctan(x) for all x∈ R
• tan(arctan x)=x , for all real numbers
• arctan(1/x)=π/2 – arctan(x) = arccot(x) ; if x>0

                                       (OR)

• arctan(1/x)=-π/2 – arctan(x) = arccot(x) -π ; if x<0
• sin(arctan x)= x/ √(1+x²)
• cos(arctan x)=1/ √(1+x²)
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