When Should I Use atan() vs. atan2() in C ?

The Distinction Between atan and atan2 in C

In mathematics, the tangent of an angle α is defined as the ratio of its sine to cosine:

tan(α) = sin(α) / cos(α)

However, this formula cannot distinguish between angles in different quadrants. To resolve this ambiguity, C provides two functions: atan and atan2.

atan: Ambiguous Angle Calculation

The atan() function returns an angle between -π/2 and π/2, regardless of the quadrant from which the original tangent value was derived. This means that atan() can only accurately represent angles in the first and fourth quadrants (where the tangent is positive).

atan2: Precise Angle Determination

Unlike atan(), the atan2() function takes two arguments: y and x. These represent the sine and cosine components of an angle, respectively. atan2() uses these values to calculate the angle, resolving all four quadrants by adding π to the result of atan() whenever the cosine is negative.

Representation of Vectors

The atan2(y, x) function is particularly useful for representing vectors. The y and x arguments represent the projection of a vector with length v and angle α on the y- and x-axis, respectively:

y = v * sin(α)
x = v * cos(α)

Thus, the relationship between these values is:

y/x = tan(α)

Conclusion

The atan() function is suitable for situations where only angles from the first or fourth quadrants are relevant. However, when precise angle determination is essential, the atan2() function should be preferred. It provides the full range of angles and can resolve the correct angle even if the input values come from different quadrants.

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