Arctangent, written as arctan or tan-1 (not khổng lồ be confused with
Using special angles khổng lồ 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 và arctangent values may be worth memorizing. Below is a table showing these angles (θ) in both radians & degrees, and their respective tangent values, tan(θ).
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To find tan(θ), we either need khổng lồ just memorize the values, or remember that tan(θ)=
Once we"ve memorized the values, or if we have a reference of some sort, it becomes relatively simple lớn recognize & determine tangent or arctangent values for the special angles.
Find arctan() và arctan(-1).
Generally, functions và their inverses exhibit the relationship
f(f-1(x)) = x và f-1(f(x)) = x
Given that x is in the domain name of the function. The same is true of tan(x) and arctan(x) within their respective restricted domains:
tan(arctan(x)) = x, for all x
arctan(tan(x)) = x, for all x in (,
These properties allow us to 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.
Composition of other trigonometric functions
We can also make compositions using all the other trigonometric functions: sine, cosine, cosecant, secant, and cotangent.
Since is not one of the ratios for the special angles, we can use a right triangle khổng lồ find the value of this composition. Given arctan() = θ, we can find that tan(θ) = . The right triangle below shows θ & the ratio of its opposite side to 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
We know that arctan() = θ, so we can rewrite the problem and find sec(θ) by using the triangle we constructed above & the fact that sec(θ) = :
sec(arctan()) = sec(θ) =
The same process can be used with a variable expression.
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Given arctan(3x) = θ, we can find that tan(θ) =
To find sine, we need khổng lồ find the hypotenuse since sin(θ)=
(3x)2 + 12 = c2
9x2 + 1 = c2
sin(arctan(3x)) = sin(θ) =
Using arctan khổng lồ solve trigonometric equations
Arctangent can also be used khổng lồ solve trigonometric equations involving the tangent function.
Solve the following trigonometric equations for x where 0≤x
2. Tan2(x) - tan(x) = 0