Inverse trigonometric

Correspondingly, the parameter 1/8q tends to zero and we can expand the inverse trigonometric function tan 1/8q in the power series... 

Derivatives of inverse trigonometric functions can be obtained in a similar fashion. For instance, the derivative of... 

More complicated equations can be solved using the inverse trigonometric functions. Suppose that... 

As before, the appropriate inverse trigonometric function is used to give... 

The remaining inverse trigonometrical functions may be left as an exercise for the student. Their values will be found on page 193. 

IV. Expressions involving the square root of a quadratic binomial can very often be readily solved by the aid of a lucky trigonometrical substitution. The form of the inverse trigonometrical functions (Table II.) will sometimes serve as a guide in the right choice. If the binomial has the forms ... 

The elliptic functions are thus related to the elliptic integrals the same as the trigonometrical functions are related to the inverse trigonometrical functions, for, as we have seen, if... 

In the following formulas u, v, w represent functions of x, while a, c, n represent fixed real numbers. All arguments in the trigonometric functions are measured in radians, and all inverse trigonometric and hyperbolic functions represent principal values Let y = f(x) and = f (x) define, respectively, a function and its derivative for any value x in their common domain. 

Now, however, in the range of values chosen for 6, cos0 is always positive. Thus we may remove the absolute value signs from cos 6 in the denominator. (This is one of the reasons that the principal values of the inverse trigonometric functions are defined as they are.) Then the cos0 terms cancel, and the integral becomes... 

All angles are measured in radians, and inverse trigonometric and hyperbolic functions represent principal values, unless otherwise indicated. 

See the section on inverse trigonometric functions for a full treatment of the allowable substitutions. 

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