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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
Note that the arctan functions implemented in computer languages only produce results between −π/2 and π/2, which is why atan2 is used to generate all the correct orientations. Moreover, typical implementations of arctan also might have some numerical disadvantages near zero and one. Some implementations use the equivalent expression: [3]
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 of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent.
The inverse tangent denoted in φ = arctan y / x must be suitably defined, taking into account the correct quadrant of (x, y), as done in the equations above. See the article on atan2 .
So, raw HTML should normally not be used for new content. However, raw HTML is still present in many mathematical articles. It is generally a good practice to convert it to {} format, but coherency must be respected; that is, such a conversion must be done in a whole article, or at least in a whole section. Moreover, such a conversion must be ...
The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc. [1] (This convention is used throughout this article.) This notation arises from the following geometric relationships: [ citation needed ] when measuring in radians, an angle of θ radians will correspond to an arc ...
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} )} .
Trigonometric functions specify the relationships between side lengths and interior angles of a right triangle. For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse.