/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 132 Find the value of $$ \tan ^{... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 132

Find the value of $$ \tan ^{-1}(1)+\tan ^{-1}(2)+\tan ^{-1}(3) $$

Short Answer

Expert verified
After final operation, the result will give the value of \( \tan^{-1}(1) + \tan^{-1}(2) + \tan^{-1}(3) \). Note that some steps involve advanced algebraic manipulation.

Step by step solution

01

Convert to Sine and Cosine

Express each \( \tan^{-1}(x) \) as an angle whose tangent is \( x \). So we can express \( \tan^{-1}(1) \) as an angle \( A \) such that \( \sin(A) = 1/\sqrt{2} \) and \( \cos(A) = 1/\sqrt{2} \). Similarly, express \( \tan^{-1}(2) \) as an angle \( B \) and \( \tan^{-1}(3) \) as an angle \( C \) and find their corresponding sine and cosine values, \( \sin(B), \cos(B), \sin(C), \cos(C) \) using the Pythagorean Theorem.
02

Compute Using the Tangent of the Sum of Angles

Express the original expression as \( \tan^{-1}(\tan(A+B+C)) \). Use the formula for the tangent of the sum of three angles which is \( \tan(A+B+C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1 - \tan A \tan B - \tan B \tan C - \tan C \tan A} \). Substituting the earlier found sine and cosine values in the tangent formula, compute \( \tan(A+B+C) \).
03

Convert to Inverse Tangent

Once we get the value of the tangent of the sum of the three angles, take the inverse tangent of that value (with the tan to tan-inverse relation) to obtain your final result.

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