Ch. 6 - Inverse Circular Functions and Trigonometric Equations

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Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 6.49

Capitolo 7, Problema 6.49

Solve each equation for exact solutions.
tan⁻¹ x - tan⁻¹ (1/x ) = π/6

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Recognize that the equation involves the difference of inverse tangent functions: $\tan^{-1} x - \tan^{-1} \left( \frac{1}{x} \right) = \frac{\pi}{6}$.

Recall the formula for the difference of inverse tangents: $\tan^{-1} a - \tan^{-1} b = \tan^{-1} \left( \frac{a - b}{1 + ab} \right)$, valid when $ab > -1$ and the angles are in the principal range.

Apply this formula with $a = x$ and $b = \frac{1}{x}$ to rewrite the left side as $\tan^{-1} \left( \frac{x - \frac{1}{x}}{1 + x \cdot \frac{1}{x}} \right) = \tan^{-1} \left( \frac{x - \frac{1}{x}}{1 + 1} \right) = \tan^{-1} \left( \frac{x - \frac{1}{x}}{2} \right)$.

Set the expression inside the inverse tangent equal to $\tan \left( \frac{\pi}{6} \right)$, since $\tan^{-1} (\text{expression}) = \frac{\pi}{6}$ implies $\text{expression} = \tan \left( \frac{\pi}{6} \right)$.

Solve the resulting equation $\frac{x - \frac{1}{x}}{2} = \tan \left( \frac{\pi}{6} \right)$ for $x$, which will lead to a quadratic equation. Then find the exact values of $x$ that satisfy the original equation.

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Inverse Tangent Function (arctan)

The inverse tangent function, denoted as tan⁻¹(x) or arctan(x), returns the angle whose tangent is x. It maps real numbers to angles typically in the range (-π/2, π/2). Understanding its properties is essential for solving equations involving arctan expressions.

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Inverse Tangent

Tangent Difference Identity for Inverse Tangents

The difference of two inverse tangents can be expressed using the formula: tan⁻¹(a) - tan⁻¹(b) = tan⁻¹((a - b) / (1 + ab)), provided the denominator is not zero. This identity helps simplify and solve equations involving differences of arctan terms.

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Inverse Tangent

Solving Trigonometric Equations for Exact Values

Solving trigonometric equations involves manipulating expressions to isolate the variable and using known angle values or identities to find exact solutions. Recognizing special angles like π/6 and their tangent values aids in determining precise answers.

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How to Solve Linear Trigonometric Equations

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