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The two figures below show 3D views of respectively atan2(y, x) and arctan( y / x ) over a region of the plane. Note that for atan2(y, x), rays in the X/Y-plane emanating from the origin have constant values, but for arctan( y / x ) lines in the X/Y-plane passing through the origin have constant
In recent literature the arctangent series is sometimes called the Mādhava–Gregory series to recognize Mādhava's priority (see also Mādhava series). [3] The special case of the arctangent of is traditionally called the Leibniz formula for π, or recently sometimes the Mādhava–Leibniz formula:
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question. The azimuthal angle is denoted by φ ∈ [ 0 , 2 π ] {\displaystyle \varphi \in [0,2\pi ]} : it is the angle between the x -axis and the projection of the radial vector onto the xy -plane.
Since the arctangent of one has a very slow convergence rate if we find two complex numbers that when multiplied will result in the same real and imaginary part we will have a Machin-like formula. An example is ( 2 + i ) {\textstyle (2+\mathrm {i} )} and ( 3 + i ) {\textstyle (3+\mathrm {i} )} .
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also arctan of same x values T, T1 or T3: arctan(x) tangent: 0.1 to 1.0: arctan(0.1) to arctan(1.0) 5.71° to 45° increase: used with C or D. T: arctan(x) tangent: 1.0 to 10.0: arctan(1.0) to arctan(10) 45° to 84.3° increase: Used with CI or DI. Also with reverse angles in red for cotangent. T2: arctan(x) tangent: 1.0 to 10.0: arctan(1.0) to ...