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Ch. 2 - Acute Angles and Right Triangles
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. 2 - Acute Angles and Right TrianglesProblema 14
Capitolo 3, Problema 14

Find exact values of the six trigonometric functions for each angle. Do not use a calculator. Rationalize denominators when applicable. 120°

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1
Recognize that 120° is in the second quadrant, where sine is positive and cosine is negative.
Express 120° as 180° - 60°, so use the reference angle 60° to find the trigonometric values.
Use the known exact values for 60°: \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), and \(\tan 60^\circ = \sqrt{3}\).
Apply the signs for the second quadrant: \(\sin 120^\circ = \sin 60^\circ\), \(\cos 120^\circ = -\cos 60^\circ\), and \(\tan 120^\circ = -\tan 60^\circ\).
Find the reciprocal functions using the definitions: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\), then rationalize denominators if needed.

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Reference Angles and Quadrants

To find trigonometric values for angles like 120°, identify the reference angle by subtracting from 180°, giving 60°. Recognize that 120° lies in the second quadrant, where sine is positive and cosine and tangent are negative. This helps determine the sign of each function.
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Reference Angles on the Unit Circle

Exact Values of Trigonometric Functions for Special Angles

Certain angles such as 30°, 45°, and 60° have known exact trigonometric values involving square roots and fractions. For 60°, sine is √3/2, cosine is 1/2, and tangent is √3. Using these exact values avoids approximations and calculator use.
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Introduction to Trigonometric Functions

Rationalizing Denominators

When trigonometric values have denominators with square roots, rationalize by multiplying numerator and denominator by the root to eliminate it. For example, convert 1/√3 to √3/3. This is a standard practice to present answers in simplified, exact form.
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Rationalizing Denominators
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