How To Find Arctan Of A Number

Arctan

In trigonometry, arctan refers to the changed tangent function. Inverse trigonometric functions are commonly accompanied past the prefix - arc. Mathematically, we represent arctan or the changed tangent function as tan^-1 x or arctan(x). As there are a full of six trigonometric functions, similarly, there are 6 inverse trigonometric functions, namely, sin^-110, cos^-1x, tan^-1x, cosec^-110, sec^-1x, and cot^-onex.

Arctan (tan^-1ten) is non the same equally i / tan x. That ways an inverse trigonometric function is non the reciprocal of the corresponding trigonometric function. The purpose of arctan is to observe the value of an unknown angle by using the value of the tangent trigonometric ratio. Navigation, physics, and engineering make widespread utilise of the arctan office. In this article, nosotros volition learn about several aspects of tan^-1x including its domain, range, graph, and the integral every bit well every bit derivative value.

ane.	What is Arctan?
2.	Arctan Formula
three.	Arctan Identities
4.	Arctan Domain and Range
5.	Backdrop of Arctan Function
half dozen.	Arctan Graph
7.	Derivative of Arctan
eight.	Integral of Arctan
9.	FAQs on Arctan

What is Arctan?

Arctan is one of the of import changed trigonometry functions. In a correct-angled triangle, the tan of an angle determines the ratio of the perpendicular and the base, that is, "Perpendicular / Base". In dissimilarity, the arctan of the ratio "Perpendicular / Base of operations" gives us the value of the respective angle between the base and the hypotenuse. Thus, arctan is the inverse of the tan role.

If the tangent of angle θ is equal to x, that is, 10 = tan θ, and so we have θ = arctan(x). Given below are some examples that can aid united states understand how the arctan function works:

• tan(π / 2) = ∞ ⇒ arctan(∞) = π/2
• tan (π / 3) = √3 ⇒ arctan(√3) = π/iii
• tan (0) = 0 ⇒ arctan(0) = 0

Suppose we accept a right-angled triangle. Let θ be the angle whose value needs to be determined. We know that tan θ will be equal to the ratio of the perpendicular and the base. Hence, tan θ = Perpendicular / Base. To find θ we will use the arctan function every bit, θ = tan^-1[Perpendicular / Base].

Arctan Formula

As discussed above, the basic formula for the arctan is given by, arctan (Perpendicular/Base) = θ, where θ is the bending betwixt the hypotenuse and the base of a right-angled triangle. We utilize this formula for arctan to find the value of angle θ in terms of degrees or radians. We can also write this formula as θ = tan^-i[Perpendicular / Base].

Arctan Identities

In that location are several arctan formulas, arctan identities and backdrop that are helpful in solving uncomplicated as well equally complicated sums on changed trigonometry. A few of them are given beneath:

• arctan(-x) = -arctan(x), for all 10 ∈ R
• tan (arctan ten) = 10, for all existent numbers 10
• arctan (tan x) = x, for 10 ∈ (-π/2, π/2)
• arctan(one/x) = π/2 - arctan(x) = arccot(x), if ten > 0 or,
  arctan(1/ten) = - π/2 - arctan(x) = arccot(x) - π, if x < 0
• sin(arctan 10) = x / √(1 + 10²)
• cos(arctan ten) = one / √(ane + 10²)
• arctan(10) = 2arctan\(\left ( \frac{10}{1 + \sqrt{1 + x^{^{2}}}} \right )\).
• arctan(x) = \(\int_{0}^{x}\frac{one}{z^{2} + 1}dz\)

We also have certain arctan formulas for π. These are given beneath.

• π/four = 4 arctan(i/5) - arctan(ane/239)
• π/4 = arctan(1/2) + arctan(1/3)
• π/4 = 2 arctan(1/two) - arctan(1/vii)
• π/4 = 2 arctan(1/3) + arctan(1/seven)
• π/4 = 8 arctan(i/10) - iv arctan(one/515) - arctan(1/239)
• π/four = three arctan(one/iv) + arctan(ane/xx) + arctan(1/1985)

How To Utilize Arctan 10 Formula?

We tin can get an in-depth understanding of the application of the arctan formula with the help of the following examples:

Example: In the right-angled triangle ABC, if the base of the triangle is 2 units and the height of the triangle is 3 units. Find the base bending.

Solution:

To find: base angle

Using arctan formula, nosotros know,
⇒ θ = arctan(3 ÷ ii) = arctan(i.5)
⇒ θ = 56.3^°

Reply: The bending is 56.3^°.

Arctan Domain and Range

All trigonometric functions including tan (x) accept a many-to-one relation. Notwithstanding, the inverse of a office can only exist if it has a one-to-i and onto relation. For this reason, the domain of tan x must be restricted otherwise the changed cannot exist. In other words, the trigonometric function must be restricted to its principal branch every bit we desire only one value.

The domain of tan x is restricted to (-π/2, π/2). The values where cos(x) = 0 have been excluded. The range of tan (x) is all real numbers. We know that the domain and range of a trigonometric function get converted to the range and domain of the changed trigonometric office, respectively. Thus, we can say that the domain of tan^-1x is all real numbers and the range is (-π/2, π/ii). An interesting fact to notation is that nosotros can extend the arctan function to complex numbers. In such a case, the domain of arctan will exist all complex numbers.

Arctan Table

Whatever bending that is expressed in degrees can also be converted into radians. To do and then we multiply the caste value by a factor of π/180°. Furthermore, the arctan part takes a real number every bit an input and outputs the corresponding unique angle value. The table given below details the arctan bending values for some real numbers. These can as well be used while plotting the arctan graph.

ten	arctan(x) (°)	arctan(x) (rad)
-∞	-90°	-π/2
-three	-71.565°	-ane.2490
-2	-63.435°	-1.1071
-√3	-60°	-π/three
-i	-45°	-π/4
-1/√3	-xxx°	-π/6
-ane/2	-26.565°	-0.4636
0	0°	0
1/ii	26.565°	0.4636
ane/√3	30°	π/6
one	45°	π/4
√3	60°	π/3
2	63.435°	1.1071
3	71.565°	1.2490
∞	90°	π/2

Arctan 10 Properties

Given beneath are some useful arctan identities based on the properties of the arctan function. These formulas can be used to simplify complex trigonometric expressions thus, increasing the ease of attempting bug.

• tan (tan^-anex) = x, for all existent numbers x
• tan^-i10 + tan^-aney = tan^-1[(10 + y)/(i - xy)], when xy < 1
  tan^-i10 - tan^-1y = tan^-1[(x - y)/(i + xy)], when xy > -i
• We take 3 formulas for 2tan^-1ten
  2tan^-iten = sin^-1(2x / (1+x²)), when |10| ≤ 1
  2tan^-anex = cos^-1((1-x²) / (ane+x^two)), when x ≥ 0
  2tan^-onex = tan^-1(2x / (1-x²)), when -1 < ten < one
• tan^-1(-x) = -tan^-iten, for all ten ∈ R
• tan^-ane(1/x) = cot^-110, when x > 0
• tan^-1x + cot^-1x = π/2, when ten ∈ R
• tan^-1(tan ten) = 10, just when x ∈ R - {10 : x = (2n + 1) (π/ii), where north ∈ Z}
  i.e., tan^-1(tan ten) = 10 but when ten is Non an odd multiple of π/ii. Otherwise, tan^-1(tan x) is undefined.

Arctan Graph

We know that the domain of arctan is R (all real numbers) and the range is (-π/2, π/ii). To plot the arctan graph we will first decide a few values of y = arctan(x). Using the values of the special angles that are already known we get the following points on the graph:

• When 10 = ∞, y = π/2
• When ten = √three, y = π/3
• When ten = 0, y = 0
• When 10 = -√3, y = -π/iii
• When x = -∞, y = -π/two

Using these we tin can plot the graph of arctan.

Arctan Derivative

To discover the derivative of arctan we can use the following algorithm.

Let y = arctan x

Taking tan on both the sides we get,

tan y = tan(arctan ten)

From the formula, we already know that tan (arctan 10) = x

tan y = x

Now on differentiating both sides and using the concatenation rule we get,

sec²y dy/dx = i

⇒ dy/dx = 1 / sec²y

According to the trigonometric identity we have secⁱⁱy = 1 + tan²y

dy/dx = 1 / (1 + tanⁱⁱy)

On commutation,

Thus, d(arctan x) / dx = 1 / (ane + x²)

Integral of Arctan x

The integral of arctan is the antiderivative of the inverse tangent part. Integration by parts is used to evaluate the integral of arctan.

Here, f(ten) = tan^-1x, m(ten) = 1

The formula is given as ∫f(x)yard(x)dx = f(ten) ∫1000(x)dx - ∫[d(f(x))/dx × ∫yard(ten) dx] dx

On substituting the values and solving the expression we go the integral of arctan every bit,

∫tan^-1x dx = ten tan^-1x - ½ ln |1+x^two| + C

where, C is the abiding of integration.

Related Manufactures:

• Arctan Calculator
• Trigonometric Functions
• Trigonometric Chart
• Sin cos tan

Important Notes on Arctan

• Arctan can also exist written as arctan x or tan^-1x. Even so, tan^-onex is not equal to (tan x)^-1 = 1 / tan x = cot 10.
• The bones formula for arctan is given every bit θ = arctan(Perpendicular / Base).
• The derivative of arctan is d/dx(tan^-1x) = 1/(ane+x²).
• The integral of arctan is ∫tan^-1ten dx = x tan^-1ten - ½ ln |ane+x²| + C

Arctan Examples

1. Case 1: Determine the value of θ if nosotros have tan^-1(1 / √iii) = θ.

   Solution: We know that tan θ = Perpendicular / Base. At present, θ = tan^-1(Perpendicular / Base of operations). We also know that tan (π / 6) = 1/ √3. Thus, π / 6 = tan^-1(ane / √3).

   Respond: θ = π / vi

2. Example 2: Suppose we have a correct-angled triangle with the dimensions, base = 1 unit, perpendicular = 1 unit, and hypotenuse = √2 units. And so notice the value of the bending between the base and the hypotenuse.

   Solution: We know that tan θ = Perpendicular / Base. Here, θ is the angle between the base and the hypotenuse.

   Thus, tan θ = 1 / i = ane

   θ = tan^-1(1)

   θ = π / 4

   Reply: The value of the bending formed between the base and the hypotenuse is π / 4.

3. Example three: Discover the value of sin(arctan(12/5)).

   Solution: Let arctan(12/5) = A. tan A = 12/v. Now tan A = Perpendicular / Base of operations. Nosotros likewise know that sin A = Perpendicular / Hypotenuse. Using the Pythagoras Theorem, Hypotenuse² = Perpendicular ²+ Base²

   Hypotenuse = √(12² + five²) = 13

   We have sin(arctan(tanA)) = sin A = 12/13

   Answer: sin(arctan(12/5)) = 12/13

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Arctan Practice Questions

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FAQs on Arctan

What is the Arctan Function in Trigonometry?

Arctan function is the changed of the tangent part. It is normally denoted equally arctan x or tan^-110. The basic formula to make up one's mind the value of arctan is θ = tan^-1(Perpendicular / Base).

Is Arctan the Inverse of Tan?

Yeah, arctan is the inverse of tan. It can determine the value of an angle in a right triangle using the tangent part. Tan^-1x will simply be if we restrict the domain of the tangent role.

Are Arctan and Cot the Aforementioned?

Arctan and cot are non the same. The inverse of the tangent function is arctan given by tan^-onex. However, cotangent is the reciprocal of the tangent function. That is (tan x)^-1 = 1 / cot 10

What is the Formula for Arctan?

The basic arctan formula can exist given by θ = tan^-1(Perpendicular / Base of operations). Here, θ is the angle between the hypotenuse and the base of operations of a right-angled triangle.

What is the Derivative of Arctan?

The derivative of arctan can exist calculated past applying the commutation and chain rule concepts. Thus, d(arctan ten) / dx = 1 / (one + 10ⁱⁱ), x ≠ i, -i.

How to Calculate the Integral of Arctan?

Nosotros will have to use integration past parts to find the value of the integral of arctan. This value is given as ∫tan^-anex dx = x tan^-1ten - ½ ln |one+x²| + C.

What is the Arctan of Infinity?

We know that the value of tan (π/2) = sin(π/2) / cos (π/two) = i / 0 = ∞. Thus, we can say that arctan(∞) = π/2.

What is the Limit of Arctan x every bit 10 Approaches Infinity?

The value of arctan approaches π/two as x approaches infinity. Likewise, we know that tan (π/two) = ∞. So, the limit of arctan is equal to π/2 every bit x tends to infinity.

Source: https://www.cuemath.com/trigonometry/arctan/#:~:text=What%20is%20the%20Arctan%20Function,1(Perpendicular%20%2F%20Base).

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