Arctangent, written as arctan or tan-1 (not to be confused with

*
) is the inverse tangent function. Tangent only has an inverse function on a restricted domain,
arctan=degrad

Using special angles lớn find arctan

While we can find the value for arctangent for any x value in the interval <-∞, ∞>, there are certain angles that are used frequently in trigonometry (0°, 30°, 45°, 60°, 90°, and their multiples and radian equivalents) whose tangent & arctangent values may be worth memorizing. Below is a table showing these angles (θ) in both radians and degrees, & their respective tangent values, tan(θ).

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θ-90°-60°-45°-30°30°45°60°90°
tan(θ)undefined
*
-1
*
01undefined

To find tan(θ), we either need lớn just memorize the values, or remember that tan(θ)=

*
, và determine the value for tan(θ) based on the sine và cosine values, which follow a pattern that may be easier khổng lồ memorize. Refer to their respective pages khổng lồ view a method that may help with memorizing sine and cosine values.

Once we"ve memorized the values, or if we have a reference of some sort, it becomes relatively simple to recognize and determine tangent or arctangent values for the special angles.


Example:

Find arctan() and arctan(-1).

*
,
*
.

*
,
*
.


Inverse properties

Generally, functions and their inverses exhibit the relationship

f(f-1(x)) = x & f-1(f(x)) = x

Given that x is in the domain name of the function. The same is true of tan(x) & arctan(x) within their respective restricted domains:

tan(arctan(x)) = x, for all x

and

arctan(tan(x)) = x, for all x in (,

*
)

These properties allow us khổng lồ evaluate the composition of trigonometric functions.

Composition of arctangent & tangent

If x is within the domain, evaluating a composition of arctan & tan is relatively simple.


Examples:

*

*


Composition of other trigonometric functions

We can also make compositions using all the other trigonometric functions: sine, cosine, cosecant, secant, and cotangent.


Example:

Find sec(arctan()).

Since is not one of the ratios for the special angles, we can use a right triangle to lớn find the value of this composition. Given arctan() = θ, we can find that tan(θ) = . The right triangle below shows θ và the ratio of its opposite side to lớn its adjacent side.

*

To find secant, we need khổng lồ find the hypotenuse since sec(θ)=. Let c be the length of the hypotenuse. Using the Pythagorean theorem,

12 + 22 = c2

5 = c2

c =

*

We know that arctan() = θ, so we can rewrite the problem and find sec(θ) by using the triangle we constructed above và the fact that sec(θ) = :

sec(arctan()) = sec(θ) =

*


The same process can be used with a variable expression.


Example:

Find sin(arctan(3x)).

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Given arctan(3x) = θ, we can find that tan(θ) =

*
và construct the following triangle:

*

To find sine, we need to lớn find the hypotenuse since sin(θ)=

*
. Let c be the length of the hypotenuse. Using the Pythagorean theorem,

(3x)2 + 12 = c2

9x2 + 1 = c2

c =

*

and

sin(arctan(3x)) = sin(θ) =

*


Using arctan to lớn solve trigonometric equations

Arctangent can also be used to lớn solve trigonometric equations involving the tangent function.


Examples:

Solve the following trigonometric equations for x where 0≤x

2. Tan2(x) - tan(x) = 0