When Should You Use ATAN2 Instead of ATAN in C ?

ATAN vs. ATAN2 in C

In C , atan and atan2 are two trigonometric functions used to calculate the arctangent of an angle. However, they differ in the way they handle information about the angle's quadrant.

ATAN: Limited Quadrant Resolution

The atan function calculates the arctangent of the quotient of the sine and cosine of an angle, i.e.:

atan(tan(α)) = α

However, it only returns an angle in the first or fourth quadrant (i.e., -π/2 ≤ atan() ≤ π/2). This limitation arises because atan cannot determine whether the input angle came from quadrant III or IV based on the quotient alone.

ATAN2: Full Quadrant Resolution

To overcome this limitation, atan2 takes two arguments: the sine and cosine of the angle, i.e.:

atan2(sin(α), cos(α)) = α

By considering both sin(α) and cos(α), atan2 can determine the correct angle in all four quadrants. Specifically:

• If cos(α) > 0, atan2 returns an angle in quadrant I.
• If cos(α) < 0 and sin(α) ≥ 0, atan2 returns an angle in quadrant II.
• If cos(α) < 0 and sin(α) < 0, atan2 returns an angle in quadrant III.
• If cos(α) = 0 and sin(α) > 0, atan2 returns π/2.
• If cos(α) = 0 and sin(α) < 0, atan2 returns -π/2.

Conclusion

While atan is convenient for simple arctangent calculations within the first and fourth quadrants, atan2 provides a more versatile and accurate solution for calculating angles in all four quadrants based on the sine and cosine values.

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