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Ch. 5 - Trigonometric Identities
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. 5 - Trigonometric IdentitiesProblema 5.1.32
Capitolo 6, Problema 5.1.32

Find the remaining five trigonometric functions of θ.
cos θ = 1/5, θ in quadrant I

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Identify the given information: \(\cos \theta = \frac{1}{5}\) and \(\theta\) is in Quadrant I, where all trigonometric functions are positive.
Use the Pythagorean identity to find \(\sin \theta\). Recall that \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = \frac{1}{5}\) into the identity: \(\sin^2 \theta + \left(\frac{1}{5}\right)^2 = 1\).
Solve for \(\sin \theta\): \(\sin^2 \theta = 1 - \left(\frac{1}{5}\right)^2 = 1 - \frac{1}{25}\). Then, \(\sin \theta = \sqrt{1 - \frac{1}{25}}\). Since \(\theta\) is in Quadrant I, take the positive root.
Find \(\tan \theta\) using the definition \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values of \(\sin \theta\) and \(\cos \theta\) found in previous steps.
Calculate the reciprocal functions: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Use the values obtained for \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\).

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Pythagorean Identity

The Pythagorean identity states that sin²θ + cos²θ = 1. Given cos θ, this identity allows us to find sin θ by rearranging the equation to sin θ = √(1 - cos²θ). This is essential for determining the sine value when cosine is known.
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Pythagorean Identities

Signs of Trigonometric Functions in Quadrants

The sign of trigonometric functions depends on the quadrant of the angle. Since θ is in quadrant I, all six trigonometric functions (sin, cos, tan, cot, sec, csc) are positive. This information helps assign the correct sign to the calculated values.
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Quadratic Formula

Definitions of Trigonometric Functions

The six trigonometric functions are defined as ratios of sides in a right triangle or as reciprocal functions: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = sin θ / cos θ, cot θ = 1 / tan θ, sec θ = 1 / cos θ, and csc θ = 1 / sin θ. Knowing these definitions allows calculation of all functions once sin θ and cos θ are known.
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Introduction to Trigonometric Functions
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