Trigonometrical functions Inverse

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... 

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... 

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

To compound the notational confusion, sin x and cos x are used to designate inverse trigonometric functions, also written as arcsinx and arccosx, respectively. These inverse functions are related by the following correspondences ... 

The derivatives of the inverse trigonometric functions, such as fi arcsinx/ix, can be evaluated using the chain rule. If 0 x) = arcsinx, then X = sin[0(x)]. Taking didx of both sides in the last form, we find... 

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