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Ch. 1 - Trigonometric Functions
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. 1 - Trigonometric FunctionsProblema 71
Capitolo 2, Problema 71

Use identities to solve each of the following. Rationalize denominators when applicable. See Examples 5–7. Find cot θ , given that csc θ = ―1.45 and θ is in quadrant III.

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Recall the identity relating cosecant and sine: \(\csc \theta = \frac{1}{\sin \theta}\). Use this to find \(\sin \theta\) by taking the reciprocal of \(\csc \theta\).
Since \(\csc \theta = -1.45\), calculate \(\sin \theta = \frac{1}{-1.45}\). Keep this as a fraction or decimal for now without simplifying fully.
Determine the sign of \(\cos \theta\) in quadrant III. In quadrant III, both sine and cosine are negative, so \(\cos \theta < 0\).
Use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\cos \theta\). Substitute the value of \(\sin \theta\) and solve for \(\cos \theta\), taking the negative root because of the quadrant.
Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Substitute the values of \(\cos \theta\) and \(\sin \theta\) found in previous steps to express \(\cot \theta\). Then rationalize the denominator if necessary.

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Reciprocal Trigonometric Identities

Reciprocal identities relate trigonometric functions to their reciprocals, such as csc θ = 1/sin θ and cot θ = 1/tan θ. Knowing csc θ allows you to find sin θ, which is essential for determining cot θ.
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Fundamental Trigonometric Identities

Sign of Trigonometric Functions in Quadrants

The sign of trig functions depends on the quadrant of the angle. In quadrant III, both sine and cosine are negative, while tangent and cotangent are positive. This helps determine the correct sign of cot θ.
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Quadratic Formula

Pythagorean Identity and Rationalizing Denominators

The Pythagorean identity, sin²θ + cos²θ = 1, allows calculation of cosine once sine is known. Rationalizing denominators ensures the final answer is simplified and free of radicals in the denominator, as required.
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Rationalizing Denominators
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