Arctan Formula – Definition, Formula, Sample Problems

Trigonometry is a study concerned with the evaluation and demonstration of sides and angles of a right-angled triangle. The operations in trigonometry are carried out by the means of sides, angles, and trigonometric ratios. These trigonometric ratios are the values of trigonometric functions which are derived from the ratios of sides and angles of the given triangle.

Trigonometry has basic trigonometric functions and these functions have their own standard trigonometric ratio value under various angles. The basic functions to be known are sine, cosine, tangent, cotangent, cosecant, and sectant. And, the inverse of these trigonometric functions is represented with the prefix ‘arc-‘ such as arcsin, arccos, arctan, arccot, arcsec, and arccosec.

What is the arctan formula?

Tangent is a trigonometric function and in a right-angled triangle, tangent equals perpendicular by base (perpendicular/base) and gives the value of the angle. 

Arctan is a reference to the inverse function of the tangent. Symbolically, arctan is represented by tan^-1x in trigonometric equations.

Suppose, the tangent of the angle θ equals x.

Then, x =tanθ 

=> θ =tan^-1x

Let us take a right-angled triangle QPR with angle θ. Now, as we studied that tangent equals perpendicular by the base.

i.e. tan θ =p/b

And, by using the same we can determine the value of angle and arctan.

As, tan θ =p/b

=> θ =tan^-1(p/b)

List of arctan Formulas

Arctan formulas can be derived from degrees or radians. These formulas help in solving equations of inverse trigonometry.

• 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)
• arctan(x)=2arctan(\frac{x}{1+\sqrt(1+x^2)})
• arctan(x)=\int^x_0\frac{1}{z^2+1}dz

There is also some standard arctan formula for \pi. These formulas are listed below.

• 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)
• π/4 = 8 arctan(1/10) – 4 arctan(1/515) – arctan(1/239)
• π/4 = 3 arctan(1/4) + arctan(1/20) + arctan(1/1985)

As we studied above that the value of arctan can be derived by degrees or radians. So, the below given table illustrates the estimated values of arctan.

x	arctan(x) (in degree)	Arctan(x) (in radian)
-∞	-90°	-π/2
-3	-71.565°	-1.2490
-2	-63.435°	-1.1071
-√3	-60°	-π/3
-1	-45°	-π/4
-1/√3	-30°	-π/6
-1/2	-26.565°	-0.4636
0	0°	0
1/2	26.565°	0.4636
1/√3	30°	π/6
1	45°	π/4
√3	60°	π/3
2	63.435°	1.1071
3	71.565°	1.2490
∞	90°	π/2

Sample Problems

Problem 1. Evaluate tan^-1(1).

Solution:

Given 

tan^-1(1)

And, the value 1 can also be written as 

1 = tan(45°)

Now,

=>tan^-1(1) = tan^-1(45°)

=>45°

Problem 2. Evaluate tan^-1(0.577).

Solution:

Given

tan^-1(0.577)

And, the value of 0.577 can also be written as

=>0.577=tan(30°)

Now,

=>tan^-1(0.577)=tan^-1(30°)

=>30°

Problem 3. Evaluate tan^-1(1.732).

Solution:

Given

tan^-1(1.732)

And, the value of 1.732 can also be written as

=>1.732=tan(60°)

Now, 

=>tan^-1(1.732)=tan^-1(60°)

=>60°

Problem 4. Evaluate tan^-1(0).

Solution:

Given

tan^-1(0)

And, the value of 0 can also be written as 

=>0 =tan(0°)

Now,

=>tan^-1(0)=tan^-1(0°)

=>0°

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