How to Calculate Arctan with Negative

Calculating arctan (inverse tangent) with negative values requires understanding the correct quadrant rules. This guide explains how to handle negative inputs, provides a calculator, shows the formula, and includes practical examples.

What is Arctan?

The arctan function, also known as the inverse tangent function, calculates the angle whose tangent is a given number. It's the inverse operation of the tangent function, which relates the angle of a right triangle to the ratio of the opposite side to the adjacent side.

Formula: arctan(y/x) = θ

Where θ is the angle in radians or degrees, and y/x is the ratio of the opposite side to the adjacent side.

The arctan function returns values in the range of -π/2 to π/2 radians (-90° to 90°), which corresponds to the first and fourth quadrants of the unit circle.

Handling Negative Values

When calculating arctan with negative values, you need to consider the signs of both the numerator (y) and denominator (x) to determine the correct quadrant:

If x is positive and y is negative, the angle is in the fourth quadrant (negative result)
If x is negative and y is positive, the angle is in the second quadrant (positive result)
If both x and y are negative, the angle is in the third quadrant (negative result)

Important: The arctan function always returns an angle between -π/2 and π/2 radians (-90° and 90°). For angles outside this range, you need to use the atan2 function which takes both x and y coordinates.

For example, arctan(-1) returns -π/4 radians (-45°), while arctan2(-1, -1) returns -3π/4 radians (-135°).

Calculation Method

The standard method for calculating arctan with negative values involves:

Determining the signs of the numerator (y) and denominator (x)
Calculating the absolute value of the ratio y/x
Using the arctan function on the absolute ratio
Applying the quadrant rules to determine the sign of the result

Calculation Steps:

If x > 0 and y < 0: θ = -arctan(|y/x|)
If x < 0 and y > 0: θ = π - arctan(|y/x|)
If x < 0 and y < 0: θ = -π + arctan(|y/x|)

For cases where both x and y are negative, you can use the atan2 function which handles all quadrant cases automatically.

Examples

Let's look at several examples of calculating arctan with negative values:

Example 1: Positive x, Negative y

Calculate arctan(-2/3):

x = 3 (positive), y = -2 (negative)
Absolute ratio: |-2/3| = 2/3
arctan(2/3) ≈ 0.588 radians (33.69°)
Apply quadrant rule: θ = -arctan(2/3) ≈ -0.588 radians (-33.69°)

Example 2: Negative x, Positive y

Calculate arctan(4/-5):

x = -5 (negative), y = 4 (positive)
Absolute ratio: |4/-5| = 4/5
arctan(4/5) ≈ 0.6747 radians (38.66°)
Apply quadrant rule: θ = π - arctan(4/5) ≈ 2.4669 radians (141.34°)

Example 3: Negative x and y

Calculate arctan(-3/-4):

x = -4 (negative), y = -3 (negative)
Absolute ratio: |-3/-4| = 3/4
arctan(3/4) ≈ 0.6435 radians (36.87°)
Apply quadrant rule: θ = -π + arctan(3/4) ≈ -2.4981 radians (-143.13°)

Arctan Calculation Examples
y/x Ratio	Quadrant	Result (radians)	Result (degrees)
-2/3	Fourth	-0.588	-33.69°
4/-5	Second	2.4669	141.34°
-3/-4	Third	-2.4981	-143.13°

FAQ

What is the range of the arctan function?

The arctan function returns values between -π/2 and π/2 radians (-90° and 90°). For angles outside this range, use the atan2 function.

How do I calculate arctan with negative values?

Calculate the absolute value of the ratio y/x, then apply the quadrant rules based on the signs of y and x to determine the correct angle.

What's the difference between arctan and atan2?

The arctan function only works with ratios (y/x) and returns angles between -π/2 and π/2. The atan2 function takes both x and y coordinates and returns angles between -π and π.

Can I use arctan for vectors?

No, for vectors you should use the atan2 function which properly handles all quadrant cases based on the signs of both coordinates.