What is Arctangent and How to Use It?

    What is Arctangent and How to Use It?

    Arctangent, also written as arctan or tan, is one of the inverse trigonometric functions. It is used to find the angle that has a given tangent ratio. For example, if we know that the tangent of an angle is 1, we can use arctangent to find that the angle is 45 degrees.

    Arctangent can be defined as follows: if tan(y) = x, then arctan(x) = y. In other words, arctangent takes a real number x and returns the angle y whose tangent is x. The range of arctangent is from -90 degrees to 90 degrees, or from -π/2 to π/2 radians.

    Arctangent is widely used in engineering, navigation, physics, and geometry to solve problems involving right triangles, circles, and angles. For example, arctangent can be used to find the direction of a slope, the angle of elevation or depression of an object, or the angle of rotation of a wheel.

    To calculate arctangent on a calculator, we can use the shift+tan buttons. Alternatively, we can use online tools such as Arctan Calculator or Wolfram Alpha. We can also use mathematical software such as MATLAB or Python to compute arctangent.

    Here are some examples of how to use arctangent:

    • If we want to find the angle whose tangent is 0.5, we can use arctan(0.5) = 26.57 degrees or 0.4636 radians.
    • If we want to find the angle whose tangent is -2, we can use arctan(-2) = -63.43 degrees or -1.107 radians.
    • If we want to find the angle whose tangent is √3, we can use arctan(√3) = 60 degrees or π/3 radians.

    For more information about arctangent and other inverse trigonometric functions, you can visit Inverse trigonometric functions – Wikipedia or arctan(x) | inverse tangent function – RapidTables .

    Arctangent is related to the other inverse trigonometric functions, such as arcsine and arccosine, by some identities. For example, we can use the following formulas to convert between them:

    • arctan(x) = arccos(1/√(1+x))
    • arctan(x) = arcsin(x/√(1+x))
    • arctan(x) = π/2 – arctan(1/x), for x ≠ 0

    We can also use arctangent to find the angle between two vectors, or the angle of a complex number. For example, if we have two vectors u = (u1, u2) and v = (v1, v2), we can use the dot product formula to find the angle θ between them:

    cos(θ) = (u · v) / (|u| |v|)

    Then we can use arccosine or arctangent to find θ:

    θ = arccos((u · v) / (|u| |v|))


    θ = arctan((u1v2 – u2v1) / (u1v1 + u2v2))

    If we have a complex number z = x + iy, we can use the polar form z = r(cos(φ) + i sin(φ)), where r = |z| and φ = arg(z). Then we can use arctangent to find φ:

    φ = arctan(y/x)

    Note that we have to consider the quadrant of z to determine the correct value of φ.

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