Tangent function, inverse

If the arctan (inverse tangent) function is defined to give values between +90° and —90°, we would have control experiment is done with r = 0, and the control field map is subtracted from the field map generated with the delay r. 

However, since we want 0 to range from 0 to 2n radians, we must specify this range for the inverse tangent function, instead of using the principal value. If we are using a calculator that is programmed to deliver the principal value, we must decide in advance which quadrant 0 lies in and be prepared to add it or lit to the calculator result if it lies in the wrong quadrant. 

In these formulas, the tan does not necessarily represent the principal value. Instead of always employing the principal branch of the inverse tangent function, one must instead use that branch of the inverse tangent function upon which f(x) lies for any particular choice of x. (This is not an issue when the antiderivative is continuous.)... 

The cutoff frequency of the transfer function is defined to be that corresponding to the point of intersection of the asymptote and the tangent to the curve describing the inverse time constant. 

The — 1 superscript indicates an inverse function. It is not an exponent, even though exponents are written in the same position. If you need to write the reciprocal of sin(y), you should write [sin (y)] to avoid confusion. It is probably better to use the notation of Eq. (2.42) rather than that of Eq. (2.43). The other inverse trigonometric functions such as the inverse cosine and inverse tangent are defined in the same way as the arcsine function. 

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