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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblema 33
Capitolo 7, Problema 33

Evaluate each expression without using a calculator.
tan (arcsin (3/5) + arccos (5/7))

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Identify the angles involved: let \( \alpha = \arcsin\left(\frac{3}{5}\right) \) and \( \beta = \arccos\left(\frac{5}{7}\right) \). We want to find \( \tan(\alpha + \beta) \).
Recall the tangent addition formula: \[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \]. So, we need to find \( \tan \alpha \) and \( \tan \beta \).
From \( \alpha = \arcsin\left(\frac{3}{5}\right) \), use the definition of sine to find the opposite side as 3 and hypotenuse as 5. Use the Pythagorean theorem to find the adjacent side: \[ \text{adjacent} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} \]. Then, calculate \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{\sqrt{16}} \).
From \( \beta = \arccos\left(\frac{5}{7}\right) \), use the definition of cosine to find the adjacent side as 5 and hypotenuse as 7. Use the Pythagorean theorem to find the opposite side: \[ \text{opposite} = \sqrt{7^2 - 5^2} = \sqrt{49 - 25} \]. Then, calculate \( \tan \beta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{24}}{5} \).
Substitute \( \tan \alpha \) and \( \tan \beta \) into the tangent addition formula and simplify the expression to find \( \tan(\alpha + \beta) \) without using a calculator.

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Inverse Trigonometric Functions

Inverse trigonometric functions like arcsin and arccos return an angle whose sine or cosine is a given value. Understanding their ranges and how to interpret these angles is essential for evaluating expressions involving compositions of trig functions.
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Introduction to Inverse Trig Functions

Trigonometric Addition Formulas

The tangent addition formula, tan(A + B) = (tan A + tan B) / (1 - tan A tan B), allows the evaluation of the tangent of a sum of angles. Applying this formula requires finding the tangent of each individual angle first.
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Quadratic Formula

Right Triangle Relationships

Using the given sine or cosine values, one can construct right triangles to find missing sides and other trigonometric ratios like tangent. This geometric approach helps compute exact values without a calculator.
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30-60-90 Triangles
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