The arctangent of 0, written as arctan(0), equals 0 because the tangent of 0 is 0.
Here's a breakdown:
Understanding Arctangent
- Arctangent (arctan), also known as the inverse tangent (tan-1), is the inverse function of the tangent function.
- It answers the question: "What angle has a tangent of this given value?".
- So, if tan(θ) = x, then arctan(x) = θ.
Why arctan(0) = 0
- The Tangent of 0: The tangent of 0 radians (or 0 degrees) is 0. This is a fundamental trigonometric fact.
- Applying the Inverse: Because tan(0) = 0, it logically follows that arctan(0) = 0. This directly utilizes the definition of an inverse function.
Equivalencies
| Function | Value |
|---|---|
| tan(0 radians) | 0 |
| arctan(0) | 0 radians |
| arctan(0) | 0 degrees |
As the reference states, "The value of arctan 0 is 0. Since the other name of arctan is 'tan inverse', we can say that tan inverse 0 is 0. Since 0 degrees is same as 0 radians: arctan 0 in degrees is 0°." The fact that zero radians and zero degrees represent the same angle makes the result consistent regardless of the unit used.
In conclusion, arctan(0) is 0 because 0 is the angle whose tangent is 0.
