The phase of a complex number z=x+i y is often written as ph z=tan^-1(y / x). Explain why this f

step 1 If Z is equal to be A plus I atab, then the trigonometric form of the complex number Z is equal

The phase of a complex number z=x+i y is often written as ph...

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Key Concepts

Quadrant Ambiguity in Determining Angles

Since the formula tan?¹(y/x) does not consider the signs of both x and y independently, it fails to account for the actual quadrant in which the complex number lies. Correctly determining the phase requires additional consideration of the quadrant to resolve the appropriate angle, a task better handled by functions like atan2.

Periodic Nature of Trigonometric Functions

Tangent is a periodic function with period ?, which means that its inverse, arctan, will yield values that repeat every ?. This periodicity implies that a single tangent value corresponds to infinitely many angles differing by multiples of ?, leading to ambiguity when trying to uniquely define an angle.

Range of the Arctan Function

The arctan function, when applied as tan?¹(y/x), typically returns a principal value in the interval (-?/2, ?/2). This limited range means that it cannot distinguish between angles that lie in different parts of the coordinate plane, such as those in the second or third quadrants, thereby causing an inherent ambiguity in the determined phase.

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